{"id":125512,"date":"2023-09-08T19:27:00","date_gmt":"2023-09-08T13:57:00","guid":{"rendered":"https:\/\/www.kopykitab.com\/blog\/?p=125512"},"modified":"2023-09-12T20:33:57","modified_gmt":"2023-09-12T15:03:57","slug":"rd-sharma-class-10-solutions-chapter-3-exercise-3-2","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-3-exercise-3-2\/","title":{"rendered":"RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 (Updated for 2023-24)"},"content":{"rendered":"\n<p><img class=\"alignnone size-full wp-image-125553\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Class-10-Solutions-Chapter-3-Exercise3.2.jpg\" alt=\"RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2\" width=\"1200\" height=\"675\" srcset=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Class-10-Solutions-Chapter-3-Exercise3.2.jpg 1200w, https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Class-10-Solutions-Chapter-3-Exercise3.2-768x432.jpg 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><strong>RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2:&nbsp;<\/strong>This exercise evaluates your understanding of how to generate graphs of linear equations while solving systems of simultaneous linear equations in two variables. Experts at Kopykitab provide all of the <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-for-maths\/\"><strong>RD Sharma Class 10 Solutions<\/strong><\/a> by connecting students at various levels. For any reference relating to the exercise questions, the <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-3-pair-of-linear-equations-in-two-variables\/\"><strong>RD Sharma Solutions for Class 10 Maths Chapter 3<\/strong><\/a> Exercise 3.2 PDF is provided here.<\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_47_1 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"ez-toc-toggle-icon-1\"><label for=\"item-69dd1dd0eb159\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;direction:ltr;\"><svg style=\"fill: #000000;color:#000000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #000000;color:#000000\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input  type=\"checkbox\" id=\"item-69dd1dd0eb159\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-visibility-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-3-exercise-3-2\/#download-rd-sharma-class-10-solutions-chapter-3-exercise-32-free-pdf\" title=\"Download RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 Free PDF\">Download RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 Free PDF<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-3-exercise-3-2\/#access-answers-to-rd-sharma-solutions-class-10-maths-chapter-3-exercise-32-important-question-with-answers\" title=\"Access answers to RD Sharma Solutions Class 10 Maths Chapter 3 Exercise 3.2- Important Question with Answers\">Access answers to RD Sharma Solutions Class 10 Maths Chapter 3 Exercise 3.2- Important Question with Answers<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-3-exercise-3-2\/#faqs-on-rd-sharma-class-10-solutions-chapter-3-exercise-32\" title=\"FAQs on RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2\">FAQs on RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-3-exercise-3-2\/#where-can-i-download-rd-sharma-class-10-solutions-chapter-3-exercise-32-free-pdf\" title=\"Where can I download RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 free PDF?\">Where can I download RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 free PDF?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-3-exercise-3-2\/#is-it-required-to-practice-all-of-the-questions-in-rd-sharma-class-10-solutions-chapter-3-exercise-32\" title=\"Is it required to practice all of the questions in RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2?\">Is it required to practice all of the questions in RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-3-exercise-3-2\/#what-are-the-benefits-of-using-rd-sharma-class-10-solutions-chapter-3-exercise-32\" title=\"What are the benefits of using RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2?\">What are the benefits of using RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"download-rd-sharma-class-10-solutions-chapter-3-exercise-32-free-pdf\"><\/span>Download RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 Free PDF<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div id=\"example1\" style=\"text-align: justify;\">&nbsp;<\/div>\n<p style=\"text-align: justify;\"><style>\n.pdfobject-container { height: 800px;}<br \/>\n.pdfobject { border: 1px solid #666; }<br \/>\n<\/style><\/p>\n<p style=\"text-align: justify;\"><script src=\"https:\/\/www.kopykitab.com\/_utility\/js\/pdfobject.min.js\"><\/script><br><script>PDFObject.embed(\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Solution-Class-10-Chapter-3-Exercise-3.2.pdf\", \"#example1\");<\/script><\/p>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/09\/RD-Sharma-Solution-Class-10-Chapter-3-Exercise-3.2.pdf\">RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2<\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"access-answers-to-rd-sharma-solutions-class-10-maths-chapter-3-exercise-32-important-question-with-answers\"><\/span>Access answers to RD Sharma Solutions Class 10 Maths Chapter 3 Exercise 3.2- Important Question with Answers<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solve the following systems of equations graphically :<\/strong><br><strong>Question 1.<\/strong><br>x + y = 3<br>2x + 5y = 12&nbsp;<strong>(C.B.S.E. 1997)<\/strong><br><strong>Solution:<\/strong><br>x + y = 3<br>=&gt; x = 3 \u2013 y<br>Substituting some different values of y, we get the corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1724\/42437050172_a9a5e48b27_o.png\" alt=\"RD Sharma Class 10 Solutions Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"136\" height=\"76\"><br>Now plot the points on the graph and join them 2x + 5y = 12<br>2x = 12 \u2013 5y<br>\u21d2 x =&nbsp;<span id=\"MathJax-Element-1-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-1\" class=\"math\"><span id=\"MathJax-Span-2\" class=\"mrow\"><span id=\"MathJax-Span-3\" class=\"mfrac\"><span id=\"MathJax-Span-4\" class=\"mrow\"><span id=\"MathJax-Span-5\" class=\"mn\">12<\/span><span id=\"MathJax-Span-6\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-7\" class=\"mn\">5<\/span><span id=\"MathJax-Span-8\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-9\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get the corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1748\/27618027157_d2c388bc66_o.png\" alt=\"Class 10 RD Sharma Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"145\" height=\"80\"><br>Now plot the points on the graph and join them we see that these two lines intersect each other at (1, 2)<br>x = 1, y = 2<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1730\/42437049952_c3189f3ced_o.png\" alt=\"Answers Of RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"371\" height=\"259\"><\/p>\n<p><strong>Question 2.<\/strong><br>x \u2013 2y = 5<br>2x + 3y = 10&nbsp;<strong>(C.B.S.E. 1997)<\/strong><br><strong>Solution:<\/strong><br>x \u2013 2y = 5 =&gt; x = 5 + 2y<br>Substituting some different values of y, we get the corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1739\/42437051942_ca1e41f286_o.png\" alt=\"RD Sharma Mathematics Class 10 Pdf Download Free Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"158\" height=\"77\"><br>Now plot the points are the graph and join them<br>2x + 3y = 10 =&gt; 2x = 10 \u2013 3y<br>\u21d2 x =&nbsp;<span id=\"MathJax-Element-2-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-10\" class=\"math\"><span id=\"MathJax-Span-11\" class=\"mrow\"><span id=\"MathJax-Span-12\" class=\"mfrac\"><span id=\"MathJax-Span-13\" class=\"mrow\"><span id=\"MathJax-Span-14\" class=\"mn\">10<\/span><span id=\"MathJax-Span-15\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-16\" class=\"mn\">3<\/span><span id=\"MathJax-Span-17\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-18\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y We get the corresponding values of x as shown below :<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1736\/42437050612_be67fa7a4a_o.png\" alt=\"RD Sharma Class 10 Maths Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"164\" height=\"73\"><br>Now plot the points on the graph and join them we see that these two lines intersect each other at (5, 0)<br>x = 5, y = 0<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/897\/42437051212_5c891d5b65_o.png\" alt=\"RD Sharma Maths Book For Class 10 Solution Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"301\" height=\"287\"><\/p>\n<p><strong>Question 3.<\/strong><br>3x + y + 1 = 0<br>2x \u2013 3y + 8 = 0&nbsp;<strong>(C.B.S.E. 1996)<\/strong><br><strong>Solution:<\/strong><br>3x + y + 1 = 0<br>y = -3x \u2013 1<br>Substituting the values of x, we get the corresponding values of y, as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/873\/42437052752_8b8c860494_o.png\" alt=\"RD Sharma 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"170\" height=\"76\"><br>Now plot the points on the graph and join them<br>2x \u2013 3y + 8 = 0<br>\u21d2 2x = 3y \u2013 8<br>\u21d2 x =&nbsp;<span id=\"MathJax-Element-3-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-19\" class=\"math\"><span id=\"MathJax-Span-20\" class=\"mrow\"><span id=\"MathJax-Span-21\" class=\"mfrac\"><span id=\"MathJax-Span-22\" class=\"mrow\"><span id=\"MathJax-Span-23\" class=\"mn\">3<\/span><span id=\"MathJax-Span-24\" class=\"mi\">y<\/span><span id=\"MathJax-Span-25\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-26\" class=\"mn\">8<\/span><\/span><span id=\"MathJax-Span-27\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get the corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1743\/27618028787_0b72aa088f_o.png\" alt=\"Solution Of RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"159\" height=\"74\"><br>Now plot the points on the graph and join then we see that these two lines intersect, each other at (-1, -2)<br>x = -1, y = 2<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1757\/27618029157_8ebbf2ef2b_o.png\" alt=\"RD Sharma Class 10 Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"294\" height=\"341\"><\/p>\n<p><strong>Question 4.<\/strong><br>2x + y \u2013 3 = 0<br>2x \u2013 3y \u2013 7 = 0&nbsp;<strong>(C.B.S.E. 1996)<\/strong><br><strong>Solution:<\/strong><br>2x + y \u2013 3 = 0 =&gt; y = -2x + 3<br>Substituting some different values of x, we get the corresponding values of y as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1732\/42437054162_64aa0195ae_o.png\" alt=\"RD Sharma 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"164\" height=\"74\"><br>Now plot the points and join them 2x \u2013 3y \u2013 7 = 0<br>2x = 3y +7<br>x =&nbsp;<span id=\"MathJax-Element-4-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-28\" class=\"math\"><span id=\"MathJax-Span-29\" class=\"mrow\"><span id=\"MathJax-Span-30\" class=\"mfrac\"><span id=\"MathJax-Span-31\" class=\"mrow\"><span id=\"MathJax-Span-32\" class=\"mn\">3<\/span><span id=\"MathJax-Span-33\" class=\"mi\">y<\/span><span id=\"MathJax-Span-34\" class=\"mo\">+<\/span><span id=\"MathJax-Span-35\" class=\"mn\">7<\/span><\/span><span id=\"MathJax-Span-36\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1753\/42437053042_93a47c7df3_o.png\" alt=\"10th Maths Solution Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"162\" height=\"78\"><br>Now plot the points on the graph and join them we see that these two lines intersect each other at (2, -1)<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1735\/27618030167_d25480307c_o.png\" alt=\"Maths RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"320\" height=\"343\"><\/p>\n<p><strong>Question 5.<\/strong><br>x + y = 6<br>x \u2013 y = 2&nbsp;<strong>(C.B.S.E. 1994)<\/strong><br><strong>Solution:<\/strong><br>x + y = 6 =&gt; x = 6 \u2013 y<br>Substituting some different values of y, we get the corresponding values of x as shown under<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1753\/27618030807_bba4944dae_o.png\" alt=\"RD Sharma Class 10 Textbook PDF Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"156\" height=\"78\"><br>Now plot the points on the graph and join them<br>x \u2013 y = 2 \u21d2 x = 2 + y<br>Substituting some different values of y, we get the corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1732\/27618030347_f18cab6d15_o.png\" alt=\"RD Sharma 10 Class Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"157\" height=\"80\"><br>Now plot the points on the graph and join them<br>We see that there two lines intersect each other at (4, 2)<br>x = 4, y = 2<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1759\/42437054622_ec6e6fbed2_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"320\" height=\"253\"><\/p>\n<p><strong>Question 6.<\/strong><br>x \u2013 2y = 6<br>3x \u2013 6y = 0&nbsp;<strong>(C.B.S.E. 1995)<\/strong><br><strong>Solution:<\/strong><br>x \u2013 2y = 6<br>x = 6 + 2 y<br>Substituting some different values ofy, we get the corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1757\/42437055852_f58b010018_o.png\" alt=\"Class 10 RD Sharma Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"171\" height=\"77\"><br>Now plot the points and join them<br>3x \u2013 6y = 0 \u21d2 3x = 6y \u21d2 x = 2y<br>Substituting some different values of y, we get corresponding the values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1741\/27618030957_6367785fa8_o.png\" alt=\"RD Sharma Class 10 Book Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"150\" height=\"83\"><br>plot the points on the graph and join them We see that these two lines intersect each other at no point<br>The lines are parallel<br>There is no solution<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1741\/27618031407_5c64fe8bd7_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"332\" height=\"292\"><\/p>\n<p><strong>Question 7.<\/strong><br>x + y = 4<br>2x \u2013 3y = 3&nbsp;<strong>(C.B.S.E. 1995)<\/strong><br><strong>Solution:<\/strong><br>x + y = 4 =&gt; y = 4 \u2013 x<br>Substituting some different values of y, we get the corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/889\/42437056922_be86bff084_o.png\" alt=\"RD Sharma Class 10 Pdf Ebook Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"151\" height=\"75\"><br>Now plot the points and join them 2x \u2013 3y = 3<br>\u21d2 2x = 3 + 3y<br>\u21d2 x =&nbsp;<span id=\"MathJax-Element-5-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-37\" class=\"math\"><span id=\"MathJax-Span-38\" class=\"mrow\"><span id=\"MathJax-Span-39\" class=\"mfrac\"><span id=\"MathJax-Span-40\" class=\"mrow\"><span id=\"MathJax-Span-41\" class=\"mn\">3<\/span><span id=\"MathJax-Span-42\" class=\"mo\">+<\/span><span id=\"MathJax-Span-43\" class=\"mn\">3<\/span><span id=\"MathJax-Span-44\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-45\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get the corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/882\/42437056272_60454ae392_o.png\" alt=\"RD Sharma Class 10 Solution Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"161\" height=\"77\"><br>Now plot the points on the graph and join them we see that these two lines intersect each other at (3, 1)<br>x = 3, y = 1<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1756\/27618032137_6869626c3b_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"322\" height=\"312\"><\/p>\n<p><strong>Question 8.<\/strong><br>2x + 3y= 4<br>x \u2013 y + 3 = 0&nbsp;<strong>(C.B.S.E. 1995)<\/strong><br><strong>Solution:<\/strong><br>2x + 3y = 4<br>=&gt; 2x = 4 \u2013 3y<br>=&gt; x =&nbsp;<span id=\"MathJax-Element-6-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-46\" class=\"math\"><span id=\"MathJax-Span-47\" class=\"mrow\"><span id=\"MathJax-Span-48\" class=\"mfrac\"><span id=\"MathJax-Span-49\" class=\"mrow\"><span id=\"MathJax-Span-50\" class=\"mn\">4<\/span><span id=\"MathJax-Span-51\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-52\" class=\"mn\">3<\/span><span id=\"MathJax-Span-53\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-54\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1739\/42437057532_c4113ebc00_o.png\" alt=\"Class 10 RD Sharma Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"159\" height=\"77\"><br>Now plot the points are the graph and join them<br>x \u2013 y + 3 = 0<br>x = y \u2013 3<br>Substituting some different values of y, we get corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1735\/42437057192_5824eb51f0_o.png\" alt=\"Learncbse.In Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"150\" height=\"75\"><br>Now plot the points on the graph and join them<br>We see that these two lines intersect each other at (-1, 2)<br>x = -1, y = 2<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1751\/27618032507_f84aa18b70_o.png\" alt=\"RD Sharma Solutions Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"276\" height=\"266\"><\/p>\n<p><strong>Question 9.<\/strong><br>2x \u2013 3y + 13 =0<br>3x \u2013 2y + 12 = 0&nbsp;<strong>(C.B.S.E. 2001C)<\/strong><br><strong>Solution:<\/strong><br>2x \u2013 3y + 13 = 0<br>2x = 3y \u2013 13<br>=&gt; x =&nbsp;<span id=\"MathJax-Element-7-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-55\" class=\"math\"><span id=\"MathJax-Span-56\" class=\"mrow\"><span id=\"MathJax-Span-57\" class=\"mfrac\"><span id=\"MathJax-Span-58\" class=\"mrow\"><span id=\"MathJax-Span-59\" class=\"mn\">3<\/span><span id=\"MathJax-Span-60\" class=\"mi\">y<\/span><span id=\"MathJax-Span-61\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-62\" class=\"mn\">13<\/span><\/span><span id=\"MathJax-Span-63\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/877\/27618033037_c2d9ae3eef_o.png\" alt=\"RD Sharma Class 10 Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"149\" height=\"73\"><br>Plot the points on the graph and join them 3x \u2013 2y + 12 = 0<br>3x = 2y \u2013 12<br>x =&nbsp;<span id=\"MathJax-Element-8-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-64\" class=\"math\"><span id=\"MathJax-Span-65\" class=\"mrow\"><span id=\"MathJax-Span-66\" class=\"mfrac\"><span id=\"MathJax-Span-67\" class=\"mrow\"><span id=\"MathJax-Span-68\" class=\"mn\">2<\/span><span id=\"MathJax-Span-69\" class=\"mi\">y<\/span><span id=\"MathJax-Span-70\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-71\" class=\"mn\">12<\/span><\/span><span id=\"MathJax-Span-72\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1734\/42437057662_acbf70e08e_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"158\" height=\"75\"><br>Plot the points and join them We see that these two lines intersect each other at (-2, 3)<br>x = -2, y = 3<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1732\/42437058212_7e8c6e4359_o.png\" alt=\"RD Sharma Class 10 Solutions Pair Of Linear Equations In Two Variables\" width=\"322\" height=\"296\"><\/p>\n<p><strong>Question 10.<\/strong><br>2x + 3y + 5 = 0<br>3x \u2013 2y \u2013 12 = 0&nbsp;<strong>(C.B.S.E. 2001 C)<\/strong><br><strong>Solution:<\/strong><br>2x + 3y + 5 = 0<br>2x = \u2013 3y \u2013 5<br>x =&nbsp;<span id=\"MathJax-Element-9-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-73\" class=\"math\"><span id=\"MathJax-Span-74\" class=\"mrow\"><span id=\"MathJax-Span-75\" class=\"mfrac\"><span id=\"MathJax-Span-76\" class=\"mrow\"><span id=\"MathJax-Span-77\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-78\" class=\"mn\">3<\/span><span id=\"MathJax-Span-79\" class=\"mi\">y<\/span><span id=\"MathJax-Span-80\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-81\" class=\"mn\">5<\/span><\/span><span id=\"MathJax-Span-82\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1741\/42437059882_eba61707df_o.png\" alt=\"Pair Of Linear Equations In Two Variables Class 10 RD Sharma\" width=\"162\" height=\"74\"><br>Now plot the points on the graph and join them<br>3x \u2013 2y \u2013 12 = 0<br>3x = 2y +12<br>x =&nbsp;<span id=\"MathJax-Element-10-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-83\" class=\"math\"><span id=\"MathJax-Span-84\" class=\"mrow\"><span id=\"MathJax-Span-85\" class=\"mfrac\"><span id=\"MathJax-Span-86\" class=\"mrow\"><span id=\"MathJax-Span-87\" class=\"mn\">2<\/span><span id=\"MathJax-Span-88\" class=\"mi\">y<\/span><span id=\"MathJax-Span-89\" class=\"mo\">+<\/span><span id=\"MathJax-Span-90\" class=\"mn\">12<\/span><\/span><span id=\"MathJax-Span-91\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/878\/27618033097_6a6949c11e_o.png\" alt=\"RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"162\" height=\"79\"><br>Now plot the points on the graph and join them we see that these lines intersect each other at (2, -3)<br>x = 2, y = -3<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1751\/27618034057_2cc57c1149_o.png\" alt=\"RD Sharma Class 10 Solutions Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"355\" height=\"322\"><br><strong>Show graphically that each one of the following systems of equations has infinitely many solutions :<\/strong><\/p>\n<p><strong>Question 11.<\/strong><br>2x + 3y = 6<br>4x + 6y = 12&nbsp;<strong>[CBSE2010]<\/strong><br><strong>Solution:<\/strong><br>2x + 3y = 6 \u2026\u2026\u2026.(i)<br>4x + 6y = 12 \u2026\u2026\u2026.(ii)<br>2x = 6 \u2013 3y<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1728\/42437060632_4338ffe26c_o.png\" alt=\"Class 10 RD Sharma Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"161\" height=\"308\"><br>Now plot the points of both lines on the graph and join them, we see that all the points lie on the same straight line<br>This system has infinitely many solutions<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/882\/27618034507_72e1f3bbaa_o.png\" alt=\"Answers Of RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"343\" height=\"292\"><\/p>\n<p><strong>Question 12.<\/strong><br>x \u2013 2y = 5<br>3x \u2013 6y = 15<br><strong>Solution:<\/strong><br>x \u2013 2y = 5<br>x = 5 + 2y<br>Substituting some different values of y, we get corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1730\/42437061172_0b76d1ac0e_o.png\" alt=\"RD Sharma Mathematics Class 10 Pdf Download Free Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"162\" height=\"73\"><br>Now plot these points on the graph and join them<br>3x \u2013 6y = 15<br>=&gt; 3x = 15 + 6y<br>x =&nbsp;<span id=\"MathJax-Element-11-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-92\" class=\"math\"><span id=\"MathJax-Span-93\" class=\"mrow\"><span id=\"MathJax-Span-94\" class=\"mfrac\"><span id=\"MathJax-Span-95\" class=\"mrow\"><span id=\"MathJax-Span-96\" class=\"mn\">15<\/span><span id=\"MathJax-Span-97\" class=\"mo\">+<\/span><span id=\"MathJax-Span-98\" class=\"mn\">6<\/span><span id=\"MathJax-Span-99\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-100\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1754\/27618034597_f48ecd1670_o.png\" alt=\"RD Sharma Class 10 Maths Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"162\" height=\"83\"><br>Now plot there points on the graph and join then<br>We see that these two lines coincide with each other<br>This system has infinitely many solutions.<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/878\/27618034817_7400c3087a_o.png\" alt=\"RD Sharma Maths Book For Class 10 Solution Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"337\" height=\"263\"><\/p>\n<p><strong>Question 13.<\/strong><br>3x +y = 8<br>6x + 2y = 16<br><strong>Solution:<\/strong><br>3x + y = 8 =&gt; y = 8 \u2013 3x<br>Substituting some different values of x, we get corresponding values of y as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1735\/27618035397_87fcf2cec5_o.png\" alt=\"RD Sharma 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"160\" height=\"74\"><br>Now plot these points on the graph and join them<br>6x + 2y \u2013 16 =&gt; 6x = 16 \u2013 2y<br>x =&nbsp;<span id=\"MathJax-Element-12-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-101\" class=\"math\"><span id=\"MathJax-Span-102\" class=\"mrow\"><span id=\"MathJax-Span-103\" class=\"mfrac\"><span id=\"MathJax-Span-104\" class=\"mrow\"><span id=\"MathJax-Span-105\" class=\"mn\">16<\/span><span id=\"MathJax-Span-106\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-107\" class=\"mn\">2<\/span><span id=\"MathJax-Span-108\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-109\" class=\"mn\">6<\/span><\/span><\/span><\/span><\/span><br>x =&nbsp;<span id=\"MathJax-Element-13-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-110\" class=\"math\"><span id=\"MathJax-Span-111\" class=\"mrow\"><span id=\"MathJax-Span-112\" class=\"mfrac\"><span id=\"MathJax-Span-113\" class=\"mrow\"><span id=\"MathJax-Span-114\" class=\"mn\">8<\/span><span id=\"MathJax-Span-115\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-116\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-117\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span>&nbsp;(Dividing by 2)<br>Substituting some different values of y, we get their corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1734\/42437061472_2910a0a5d1_o.png\" alt=\"Solution Of RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"148\" height=\"74\"><br>Now plot the points and point them<br>We see that the two lines coincide with each other<br>This system has infinitely many solutions<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1753\/42437062062_42b4fcc93c_o.png\" alt=\"RD Sharma Class 10 Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"248\" height=\"368\"><\/p>\n<p><strong>Question 14.<\/strong><br>x- 2y + 11 = 0<br>3x \u2013 6y + 33 = 0<br><strong>Solution:<\/strong><br>x \u2013 2y + 11 = 0<br>x = 2y \u2013 11<br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1728\/27618035927_f764fb378f_o.png\" alt=\"RD Sharma 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"158\" height=\"73\"><br>Plot the points on the graph and join them 3x \u2013 6y + 33 = 0<br>3x = 6y \u2013 33<br>x =&nbsp;<span id=\"MathJax-Element-14-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-118\" class=\"math\"><span id=\"MathJax-Span-119\" class=\"mrow\"><span id=\"MathJax-Span-120\" class=\"mfrac\"><span id=\"MathJax-Span-121\" class=\"mrow\"><span id=\"MathJax-Span-122\" class=\"mn\">6<\/span><span id=\"MathJax-Span-123\" class=\"mi\">y<\/span><span id=\"MathJax-Span-124\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-125\" class=\"mn\">33<\/span><\/span><span id=\"MathJax-Span-126\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1759\/27618035567_66e2707b39_o.png\" alt=\"RD Sharma 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"142\" height=\"76\"><br>Plot the points on the graph and join them we see that the two lines coincide with each other<br>This system has infinitely many solutions.<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1725\/27618035857_246fd8cb88_o.png\" alt=\"10th Maths Solution Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"360\" height=\"335\"><br><strong>Show graphically that each one of the following systems of equations is inconsistent (i.e., has no solution)<\/strong><\/p>\n<p><strong>Question 15.<\/strong><br>3x \u2013 5y = 20<br>6x \u2013 10y = -40&nbsp;<strong>(C.B.S.E. 1995C)<\/strong><br><strong>Solution:<\/strong><br>3x \u2013 5y = 20<br>3x = 20 + 5y<br>x =&nbsp;<span id=\"MathJax-Element-15-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-127\" class=\"math\"><span id=\"MathJax-Span-128\" class=\"mrow\"><span id=\"MathJax-Span-129\" class=\"mfrac\"><span id=\"MathJax-Span-130\" class=\"mrow\"><span id=\"MathJax-Span-131\" class=\"mn\">20<\/span><span id=\"MathJax-Span-132\" class=\"mo\">+<\/span><span id=\"MathJax-Span-133\" class=\"mn\">5<\/span><span id=\"MathJax-Span-134\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-135\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1745\/27618036537_f109906657_o.png\" alt=\"Maths RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"170\" height=\"76\"><br>Plot the points on the graph and join them<br>6x \u2013 10y = -40<br>6x = 10y \u2013 40<br>x =&nbsp;<span id=\"MathJax-Element-16-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-136\" class=\"math\"><span id=\"MathJax-Span-137\" class=\"mrow\"><span id=\"MathJax-Span-138\" class=\"mfrac\"><span id=\"MathJax-Span-139\" class=\"mrow\"><span id=\"MathJax-Span-140\" class=\"mn\">10<\/span><span id=\"MathJax-Span-141\" class=\"mi\">y<\/span><span id=\"MathJax-Span-142\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-143\" class=\"mn\">40<\/span><\/span><span id=\"MathJax-Span-144\" class=\"mn\">6<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1754\/27618036087_275d133cd3_o.png\" alt=\"Maths RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"150\" height=\"76\"><br>Plot the points on the graph and join them we see that the lines are parallel<br>The given system of equations is inconsistent and has no solution.<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1748\/27618036447_442545b1c0_o.png\" alt=\"RD Sharma Class 10 Textbook PDF Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"365\" height=\"449\"><\/p>\n<p><strong>Question 16.<\/strong><br>x \u2013 2y = 6<br>3x \u2013 6y = 0&nbsp;<strong>(C.B.S.E. 1995)<\/strong><br><strong>Solution:<\/strong><br>x \u2013 2y = 6<br>x = 6 + 2y<br>Substituting some different values of y, we get their corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/886\/42437064642_607139b512_o.png\" alt=\"RD Sharma Class 10 Textbook PDF Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"157\" height=\"72\"><br>Plot the points on the graph and join them<br>3x \u2013 6y = 0<br>=&gt; 3x = 6y<br>=&gt; x = 2y<br>Substituting some different values of y, we get their corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1733\/27618036747_3f643f6dbb_o.png\" alt=\"RD Sharma Class 10 Textbook PDF Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"151\" height=\"77\"><br>Plot the points on the graph and join them We see that the lines are parallel<br>The system of equations is inconsistent and therefore has no solution.<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1759\/41585870405_54952bcc97_o.png\" alt=\"RD Sharma 10 Class Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"306\" height=\"241\"><\/p>\n<p><strong>Question 17.<\/strong><br>2y \u2013 x = 9<br>6y \u2013 3x = 21&nbsp;<strong>(C.B.S.E. 1995C)<\/strong><br><strong>Solution:<\/strong><br>2y \u2013 x = 9<br>=&gt; x = 2y \u2013 9<br>Substituting some different values of y, we get their corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/893\/42437065842_0d3a1855a6_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"160\" height=\"77\"><br>Now plot the points on the graph and join them<br>6y \u2013 3x = 21<br>=&gt; 6y = 21 + 3x<br>y =&nbsp;<span id=\"MathJax-Element-17-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-145\" class=\"math\"><span id=\"MathJax-Span-146\" class=\"mrow\"><span id=\"MathJax-Span-147\" class=\"mfrac\"><span id=\"MathJax-Span-148\" class=\"mrow\"><span id=\"MathJax-Span-149\" class=\"mn\">21<\/span><span id=\"MathJax-Span-150\" class=\"mo\">+<\/span><span id=\"MathJax-Span-151\" class=\"mn\">3<\/span><span id=\"MathJax-Span-152\" class=\"mi\">x<\/span><\/span><span id=\"MathJax-Span-153\" class=\"mn\">6<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of x, we get their corresponding values of y as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1742\/42437065012_27effb8872_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"158\" height=\"76\"><br>Now plot the points on the graph and join them we see that the lines are parallel<br>The system of equations is inconsistent and therefore has no solution.<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/881\/42437065582_ea68dc99f2_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"346\" height=\"288\"><\/p>\n<p><strong>Question 18.<\/strong><br>3x \u2013 4y \u2013 1 = 0<br>2x \u2013&nbsp;<span id=\"MathJax-Element-18-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-154\" class=\"math\"><span id=\"MathJax-Span-155\" class=\"mrow\"><span id=\"MathJax-Span-156\" class=\"mfrac\"><span id=\"MathJax-Span-157\" class=\"mn\">8<\/span><span id=\"MathJax-Span-158\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span>&nbsp;y + 5 = 0<br><strong>Solution:<\/strong><br>3x \u2013 4y -1 = 0<br>3x = 4y + 1<br>x =&nbsp;<span id=\"MathJax-Element-19-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-159\" class=\"math\"><span id=\"MathJax-Span-160\" class=\"mrow\"><span id=\"MathJax-Span-161\" class=\"mfrac\"><span id=\"MathJax-Span-162\" class=\"mrow\"><span id=\"MathJax-Span-163\" class=\"mn\">4<\/span><span id=\"MathJax-Span-164\" class=\"mi\">y<\/span><span id=\"MathJax-Span-165\" class=\"mo\">+<\/span><span id=\"MathJax-Span-166\" class=\"mn\">1<\/span><\/span><span id=\"MathJax-Span-167\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1749\/42437067162_39f760d150_o.png\" alt=\"Class 10 RD Sharma Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"162\" height=\"72\"><br>Now plot the points on the graph and join them<br>2x \u2013&nbsp;<span id=\"MathJax-Element-20-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-168\" class=\"math\"><span id=\"MathJax-Span-169\" class=\"mrow\"><span id=\"MathJax-Span-170\" class=\"mfrac\"><span id=\"MathJax-Span-171\" class=\"mn\">8<\/span><span id=\"MathJax-Span-172\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span>&nbsp;y + 5 = 0<br>=&gt; 6x \u2013 8y + 15 = 0<br>=&gt; 6x = 8y \u2013 15<br>=&gt; x =&nbsp;<span id=\"MathJax-Element-21-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-173\" class=\"math\"><span id=\"MathJax-Span-174\" class=\"mrow\"><span id=\"MathJax-Span-175\" class=\"mfrac\"><span id=\"MathJax-Span-176\" class=\"mrow\"><span id=\"MathJax-Span-177\" class=\"mn\">8<\/span><span id=\"MathJax-Span-178\" class=\"mi\">y<\/span><span id=\"MathJax-Span-179\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-180\" class=\"mn\">15<\/span><\/span><span id=\"MathJax-Span-181\" class=\"mn\">6<\/span><\/span><\/span><\/span><\/span><br>Now substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1722\/42437066022_f47723ff8d_o.png\" alt=\"Class 10 RD Sharma Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"252\" height=\"109\"><br>Plot the points on the graph and join them We see that the lines are parallel<br>The system of equations is inconsistent Therefore has no solution.<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1727\/42437066912_30a40481e1_o.png\" alt=\"Class 10 RD Sharma Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"349\" height=\"299\"><\/p>\n<p><strong>Question 19.<\/strong><br>Determine graphically the vertices of the triangle the equations of whose sides are given below :<br>(i) 2y \u2013 x = 8, 5y \u2013 x = 14 and y \u2013 2x = 1&nbsp;<strong>(C.B.S.E. 1994)<\/strong><br>(ii) y = x, y = 0 and 3x + 3y = 10&nbsp;<strong>(C.B.S.E. 1994)<\/strong><br><strong>Solution:<\/strong><br>(i) Equations of the sides of a triangle are 2y \u2013 x = 8, 5y \u2013 x = 14 and y \u2013 2x = 1<br>2y \u2013 x = 8<br>x = 2y \u2013 8<br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1729\/28615893068_276faa9187_o.png\" alt=\"RD Sharma Class 10 Book Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"159\" height=\"68\"><br>Now plot the points and join them Similarly in 5y \u2013 x = 14<br>x = 5y \u2013 14<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1732\/41585873225_4063ca4294_o.png\" alt=\"RD Sharma Class 10 Book Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"251\" height=\"170\"><br>Now plot these points and join them in each case<br>We see that these lines intersect at (-4, 2), (1, 3), and (2, 5) which are the vertices of the triangle so formed.<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/893\/41585874005_2cfb58f85a_o.png\" alt=\"RD Sharma Class 10 Book Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"350\" height=\"270\"><br>(ii) y = x, y = 0 and 3x + 3y = 10<br>y = x<br>Substituting some different values of x, we get their corresponding values of y, as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1759\/28615892648_16fb509a33_o.png\" alt=\"RD Sharma Class 10 Book Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"355\" height=\"496\"><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1741\/41585874755_313c3a0839_o.png\" alt=\"RD Sharma Class 10 Book Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"333\" height=\"302\"><\/p>\n<p><strong>Question 20.<\/strong><br>Determine graphically whether the system of equations x \u2013 2y = 2, 4x \u2013 2y = 5 is consistent or in-consistent ?<br><strong>Solution:<\/strong><br>x \u2013 2y = 2<br>x = 2y + 2<br>Substituting some values of y, we get their corresponding values of x, as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1758\/28615893878_6d12aab06c_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"158\" height=\"77\"><br>Now plot the points on the graph and join them<br>4x \u2013 2y = 5<br>4x = 2y + 5<br>x =&nbsp;<span id=\"MathJax-Element-22-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-182\" class=\"math\"><span id=\"MathJax-Span-183\" class=\"mrow\"><span id=\"MathJax-Span-184\" class=\"mfrac\"><span id=\"MathJax-Span-185\" class=\"mrow\"><span id=\"MathJax-Span-186\" class=\"mn\">2<\/span><span id=\"MathJax-Span-187\" class=\"mi\">y<\/span><span id=\"MathJax-Span-188\" class=\"mo\">+<\/span><span id=\"MathJax-Span-189\" class=\"mn\">5<\/span><\/span><span id=\"MathJax-Span-190\" class=\"mn\">4<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values bf x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/874\/28615893328_b32cd2e117_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"185\" height=\"106\"><br>Now plot the above points and join them We see that there two lines intersect each other<br>The system is consistent<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1747\/28615893658_a2b672eaa8_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"328\" height=\"281\"><\/p>\n<p><strong>Question 21.<\/strong><br>Determine by drawing graphs, whether the following system of linear equations has a unique solution or not :<br>(i) 2x \u2013 3y = 6, x + y = 1&nbsp;<strong>(C.B.S.E. 1994)<\/strong><br>(ii) 2y = 4x \u2013 6, 2x = y + 3&nbsp;<strong>(C.B.S.E. 1995C)<\/strong><br><strong>Solution:<\/strong><br>(i) 2x \u2013 3y = 6, x + y = 1<br>2x \u2013 3y = 6<br>=&gt; 2x = 6 + 3y<br>=&gt; x =&nbsp;<span id=\"MathJax-Element-23-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-191\" class=\"math\"><span id=\"MathJax-Span-192\" class=\"mrow\"><span id=\"MathJax-Span-193\" class=\"mfrac\"><span id=\"MathJax-Span-194\" class=\"mrow\"><span id=\"MathJax-Span-195\" class=\"mn\">6<\/span><span id=\"MathJax-Span-196\" class=\"mo\">+<\/span><span id=\"MathJax-Span-197\" class=\"mn\">3<\/span><span id=\"MathJax-Span-198\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-199\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x show below<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/894\/41585877475_535bbbeb9d_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"159\" height=\"72\"><br>Now plot the points and join them<br>x + y = 1 =&gt; x = 1 \u2013 y<br>Substituting some different of y, we get their corresponding value of x as given below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1740\/28615894058_afce631164_o.png\" alt=\"RD Sharma Class 10 Pdf Ebook Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"160\" height=\"75\"><br>Now plot the points on the graph and join them we see that the lines intersect at a point<br>This system has a unique solution.<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1736\/28615894448_f34c830728_o.png\" alt=\"RD Sharma Class 10 Pdf Ebook Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"282\" height=\"309\"><br>(ii) 2y = 4x \u2013 6, 2x = y + 3<br>2y = 4x \u2013 6<br>y =&nbsp;<span id=\"MathJax-Element-24-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-200\" class=\"math\"><span id=\"MathJax-Span-201\" class=\"mrow\"><span id=\"MathJax-Span-202\" class=\"mfrac\"><span id=\"MathJax-Span-203\" class=\"mrow\"><span id=\"MathJax-Span-204\" class=\"mn\">4<\/span><span id=\"MathJax-Span-205\" class=\"mi\">x<\/span><span id=\"MathJax-Span-206\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-207\" class=\"mn\">6<\/span><\/span><span id=\"MathJax-Span-208\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;= 2x \u2013 3<br>Substituting some different values of x, we get their corresponding values of y as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1758\/28615894648_d6b57ec17e_o.png\" alt=\"RD Sharma Class 10 Pdf Ebook Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"147\" height=\"71\"><br>Now plot the points on the graph and join them<br>2x = y + 3<br>x =&nbsp;<span id=\"MathJax-Element-25-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-209\" class=\"math\"><span id=\"MathJax-Span-210\" class=\"mrow\"><span id=\"MathJax-Span-211\" class=\"mfrac\"><span id=\"MathJax-Span-212\" class=\"mrow\"><span id=\"MathJax-Span-213\" class=\"mi\">y<\/span><span id=\"MathJax-Span-214\" class=\"mo\">+<\/span><span id=\"MathJax-Span-215\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-216\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/882\/28615894808_0b509f4a69_o.png\" alt=\"RD Sharma Class 10 Pdf Ebook Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"155\" height=\"74\"><br>Plot the points on the graph and join them We see the lines coincide each other<br>This system has no unique solution.<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1752\/28615895298_e96d04d1de_o.png\" alt=\"RD Sharma Class 10 Pdf Ebook Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"255\" height=\"356\"><\/p>\n<p><strong>Question 22.<\/strong><br>Solve graphically each of the following systems of linear equations. Also And the coordinates of the points where the lines<br>meet axis of y.<br>(i) 2x \u2013 5y + 4 = 0<br>2x + y \u2013 8 = 0&nbsp;<strong>(C.B.S.E. 2005)<\/strong><br>(ii) 3x + 2y = 12<br>5x \u2013 2y = 4&nbsp;<strong>(C.B.S.E. 2000C)<\/strong><br>(iii) 2x + y \u2013 11 = 0<br>x \u2013 y \u2013 1=0&nbsp;<strong>(C.B.S.E. 2000C)<\/strong><br>(iv) x + 2y \u2013 7 = 0<br>2x \u2013 y \u2013 4 = 0&nbsp;<strong>(C.B.S.E. 2000C)<\/strong><br>(v) 3x + y \u2013 5 = 0<br>2x \u2013 y \u2013 5 = 0&nbsp;<strong>(C.B.S.E. 2002C)<\/strong><br>(vi) 2x \u2013 y \u2013 5 = 0<br>x \u2013 y \u2013 3 = 0&nbsp;<strong>(C.B.S.E. 2002C)<\/strong><br><strong>Solution:<\/strong><br>(i) 2x \u2013 5y + 4 = 0, 2x \u2013 5y + 4 = 0<br>2x \u2013 5y + 4 = 0 \u21d2 2x = 5y \u2013 4<br>\u21d2 x =&nbsp;<span id=\"MathJax-Element-26-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-217\" class=\"math\"><span id=\"MathJax-Span-218\" class=\"mrow\"><span id=\"MathJax-Span-219\" class=\"mfrac\"><span id=\"MathJax-Span-220\" class=\"mrow\"><span id=\"MathJax-Span-221\" class=\"mn\">5<\/span><span id=\"MathJax-Span-222\" class=\"mi\">y<\/span><span id=\"MathJax-Span-223\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-224\" class=\"mn\">4<\/span><\/span><span id=\"MathJax-Span-225\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown here<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1729\/41585885215_9218566352_o.png\" alt=\"RD Sharma Class 10 Solution Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"163\" height=\"74\"><br>Now plot the points on the graph and join them<br>2x + y \u2013 8 = 0 =&gt; 2x = 8 \u2013 y<br>x =&nbsp;<span id=\"MathJax-Element-27-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-226\" class=\"math\"><span id=\"MathJax-Span-227\" class=\"mrow\"><span id=\"MathJax-Span-228\" class=\"mfrac\"><span id=\"MathJax-Span-229\" class=\"mrow\"><span id=\"MathJax-Span-230\" class=\"mn\">8<\/span><span id=\"MathJax-Span-231\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-232\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-233\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1744\/41585877715_e47d790805_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"154\" height=\"75\"><br>Now join these points and join them<br>We see that the lines intersect each other at (3, 2)<br>x = 3, y = 2<br>These line intersect y-axis at(0,&nbsp;<span id=\"MathJax-Element-28-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-234\" class=\"math\"><span id=\"MathJax-Span-235\" class=\"mrow\"><span id=\"MathJax-Span-236\" class=\"mfrac\"><span id=\"MathJax-Span-237\" class=\"mn\">4<\/span><span id=\"MathJax-Span-238\" class=\"mn\">5<\/span><\/span><\/span><\/span><\/span>) and (0, 8) respectively.<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/882\/41585878985_5878316be1_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"359\" height=\"362\"><br>(ii) 3x + 2y = 12, 5x \u2013 2y = 4<br>3x + 2y = 12<br>=&gt; 3x = 12 \u2013 2y<br>x =&nbsp;<span id=\"MathJax-Element-29-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-239\" class=\"math\"><span id=\"MathJax-Span-240\" class=\"mrow\"><span id=\"MathJax-Span-241\" class=\"mfrac\"><span id=\"MathJax-Span-242\" class=\"mrow\"><span id=\"MathJax-Span-243\" class=\"mn\">12<\/span><span id=\"MathJax-Span-244\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-245\" class=\"mn\">2<\/span><span id=\"MathJax-Span-246\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-247\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1721\/28615896588_058355c07b_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"161\" height=\"75\"><br>Now plot the points and join them Similarly in 5x \u2013 2y = 4<br>=&gt; 5x = 4 + 2y<br>x =&nbsp;<span id=\"MathJax-Element-30-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-248\" class=\"math\"><span id=\"MathJax-Span-249\" class=\"mrow\"><span id=\"MathJax-Span-250\" class=\"mfrac\"><span id=\"MathJax-Span-251\" class=\"mrow\"><span id=\"MathJax-Span-252\" class=\"mn\">4<\/span><span id=\"MathJax-Span-253\" class=\"mo\">+<\/span><span id=\"MathJax-Span-254\" class=\"mn\">2<\/span><span id=\"MathJax-Span-255\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-256\" class=\"mn\">5<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1758\/42437070762_25eb2a5099_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"155\" height=\"79\"><br>Now join these points and join them<br>We sec that these lines intersect each other at (2, 3)<br>x = 2, y= 3<br>These lines intersect the y-axis at (0, 6) and (0, 2) respectively.<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/874\/42437071182_7cd1dc124e_o.png\" alt=\"Class 10 RD Sharma Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"258\" height=\"384\"><br>(iii) 2x + y \u2013 11 = 0, x \u2013 y \u2013 1 = 0<br>2x + y \u2013 11 = 0 =&gt; y = 11 \u2013 2x<br>Substituting some different values of x, we get their corresponding values of y as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/893\/28615897228_cec0984efb_o.png\" alt=\"Class 10 RD Sharma Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"163\" height=\"76\"><br>Now plot the points and join them Similarly in x \u2013 y \u2013 1= 0 =&gt; x = y + 1<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1756\/28615897498_b5b8caac73_o.png\" alt=\"Class 10 RD Sharma Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"152\" height=\"74\"><br>Now plot the points and join them We see that these two lines intersect each other at (4, 3) and intersect the y-axis at (0, 11) and (0,-1)<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/898\/28615898448_671bc7bdf7_o.png\" alt=\"Class 10 RD Sharma Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"281\" height=\"451\"><br>(iv) x + 2y \u2013 7 = 0, 2x \u2013 y \u2013 4 = 0<br>x + 2y \u2013 7 = 0<br>x = 7 \u2013 2y<br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1724\/42437072262_f1d82f04ef_o.png\" alt=\"Learncbse.In Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"152\" height=\"76\"><br>Now plot these points and join them Similarly in<br>2x \u2013 y \u2013 4 = 0<br>y = 2x \u2013 4<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/890\/28615898838_328832538f_o.png\" alt=\"Learncbse.In Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"151\" height=\"73\"><br>Now plot these points and join them We see that these two lines intersect each other at (3, 2)<br>and these lines intersect the y-axis at (0, <span id=\"MathJax-Element-31-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-257\" class=\"math\"><span id=\"MathJax-Span-258\" class=\"mrow\"><span id=\"MathJax-Span-259\" class=\"mfrac\"><span id=\"MathJax-Span-260\" class=\"mn\">7<\/span><span id=\"MathJax-Span-261\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>) and (0, -4)<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1759\/28615899608_2062986676_o.png\" alt=\"RD Sharma Solutions Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"326\" height=\"394\"><br>(v) 3x + y \u2013 5 = 0, 2x \u2013 y \u2013 5 = 0<br>3x + y \u2013 5= 0<br>y = 5 \u2013 3x<br>Substituting some different values of x, we get corresponding values of y as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/873\/42437072942_4bef6c76e9_o.png\" alt=\"RD Sharma Class 10 Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"154\" height=\"76\"><br>Now plot these points and join them Similarly in 2x \u2013 y \u2013 5 = 6 =&gt; y = 2x \u2013 5<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/893\/42437073062_07238a833f_o.png\" alt=\"RD Sharma Class 10 Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"156\" height=\"73\"><br>Now plot these points and join them We see that these two lines intersect each other at (2, -1)<br>x = 2, y = 1<br>and these Lines intersect the y-axis at (0, 5) and (0, -5) respectively.<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1747\/28615900928_44d464deba_o.png\" alt=\"RD Sharma Class 10 Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"321\" height=\"383\"><br>(vi) 2x \u2013 y \u2013 5 = 0, x \u2013 y \u2013 3 = 0<br>2x \u2013 y \u2013 5 = 0<br>y = 2x \u2013 5<br>Substituting some different values of x, we get their corresponding values of y as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1721\/41585883745_9480ed4044_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"157\" height=\"71\"><br>Plot the points and join them Similarly in-the equation x \u2013 y \u2013 3 = 0 =&gt; x =y + 3<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/890\/27618055107_4863540019_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"157\" height=\"71\"><br>Plot these points on the graph and join them we see that these two lines intersect each other at (2, -1)<br>x = 2, y = 1<br>and these lines intersect y-axis at (0, -5) and (0, -3)<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1735\/27618056207_35709262f0_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"293\" height=\"313\"><\/p>\n<p><strong>Question 23.<\/strong><br>Solve the following system of linear equations graphically and shade the region between the two lines and x-axis<br>(i) 2x + 3y = 12, x \u2013 y = 1&nbsp;<strong>(C.B.S.E. 2001)<\/strong><br>(ii) 3x + 2y \u2013 4 = 0, 2x \u2013 3y \u2013 7 = 0&nbsp;<strong>(C.B.S.E. 2006C)<\/strong><br>(iii) 3x + 2y \u2013 11 = 0, 2x \u2013 3y + 10 = 0&nbsp;<strong>(C.B.S.E. 2006C)<\/strong><br><strong>Solution:<\/strong><br>(i) 2x + 3y = 12, x \u2013 y = 1<br>2x + 3y = 12 =&gt; 2x = 12 \u2013 3y<br>=&gt; x =&nbsp;<span id=\"MathJax-Element-32-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-262\" class=\"math\"><span id=\"MathJax-Span-263\" class=\"mrow\"><span id=\"MathJax-Span-264\" class=\"mfrac\"><span id=\"MathJax-Span-265\" class=\"mrow\"><span id=\"MathJax-Span-266\" class=\"mn\">12<\/span><span id=\"MathJax-Span-267\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-268\" class=\"mn\">3<\/span><span id=\"MathJax-Span-269\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-270\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below<br><img src=\"https:\/\/farm2.staticflickr.com\/1728\/41765943714_e1c1842140_o.png\" alt=\"rd-sharma-class-10-solutions-chapter-3-pair-of-linear-equations-in-two-variables-ex-3-2-23\" width=\"149\" height=\"74\"><br>Now plot the points on the graph and join them. Similarly in the equation<br>x \u2013 y = 1 =&gt; x = 1 + y<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1732\/41585885355_7ffa2ed132_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"150\" height=\"72\"><br>Now plot the points on the graph and join them<br>We see the two lines intersect each other at (3, 2) and intersect also x-axis at (6, 0) and 0,0)<br>The required region has been shaded.<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/873\/41585885805_c1824268e8_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"293\" height=\"265\"><br>(ii) 3x + 2y \u2013 4 = 0, 2x \u2013 3y \u2013 7 = 0<br>3x + 2y \u2013 4 = 0<br>=&gt; 3x = 4 \u2013 2y<br>x =&nbsp;<span id=\"MathJax-Element-33-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-271\" class=\"math\"><span id=\"MathJax-Span-272\" class=\"mrow\"><span id=\"MathJax-Span-273\" class=\"mfrac\"><span id=\"MathJax-Span-274\" class=\"mrow\"><span id=\"MathJax-Span-275\" class=\"mn\">4<\/span><span id=\"MathJax-Span-276\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-277\" class=\"mn\">2<\/span><span id=\"MathJax-Span-278\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-279\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1747\/41585885925_9978a68ca2_o.png\" alt=\"RD Sharma Class 10 Solutions Pair Of Linear Equations In Two Variables\" width=\"152\" height=\"73\"><br>Now plot the points on the graph and join them. Similarly in the equation<br>2x \u2013 3y \u2013 7 = 0<br>=&gt; 2x = 3y + 7<br>=&gt; x =&nbsp;<span id=\"MathJax-Element-34-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-280\" class=\"math\"><span id=\"MathJax-Span-281\" class=\"mrow\"><span id=\"MathJax-Span-282\" class=\"mfrac\"><span id=\"MathJax-Span-283\" class=\"mrow\"><span id=\"MathJax-Span-284\" class=\"mn\">3<\/span><span id=\"MathJax-Span-285\" class=\"mi\">y<\/span><span id=\"MathJax-Span-286\" class=\"mo\">+<\/span><span id=\"MathJax-Span-287\" class=\"mn\">7<\/span><\/span><span id=\"MathJax-Span-288\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1758\/41585886205_3b09cce3e4_o.png\" alt=\"RD Sharma Class 10 Solutions Pair Of Linear Equations In Two Variables\" width=\"154\" height=\"78\"><br>Plot these points and join them<br>The required region is surrounded by these two lines and the x-axis has been shaded as shown.<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/901\/41765941144_4db9a4824f_o.png\" alt=\"RD Sharma Class 10 Solutions Pair Of Linear Equations In Two Variables\" width=\"323\" height=\"394\"><br>(iii) 3x + 2y \u2013 11 = 0, 2x \u2013 3y + 10 = 0<br>3x + 2y \u2013 11<br>=&gt; 3x = 11 \u2013 2y<br>=&gt; x =&nbsp;<span id=\"MathJax-Element-35-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-289\" class=\"math\"><span id=\"MathJax-Span-290\" class=\"mrow\"><span id=\"MathJax-Span-291\" class=\"mfrac\"><span id=\"MathJax-Span-292\" class=\"mrow\"><span id=\"MathJax-Span-293\" class=\"mn\">11<\/span><span id=\"MathJax-Span-294\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-295\" class=\"mn\">2<\/span><span id=\"MathJax-Span-296\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-297\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1760\/40680848200_d1951511d1_o.png\" alt=\"RD Sharma Class 10 Solutions Pair Of Linear Equations In Two Variables\" width=\"154\" height=\"74\"><br>Now plot the points and join them. Similarly in the equation<br>2x \u2013 3y + 10 = 0<br>2x = 3y \u2013 10<br>x =&nbsp;<span id=\"MathJax-Element-36-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-298\" class=\"math\"><span id=\"MathJax-Span-299\" class=\"mrow\"><span id=\"MathJax-Span-300\" class=\"mfrac\"><span id=\"MathJax-Span-301\" class=\"mrow\"><span id=\"MathJax-Span-302\" class=\"mn\">3<\/span><span id=\"MathJax-Span-303\" class=\"mi\">y<\/span><span id=\"MathJax-Span-304\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-305\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-306\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1744\/40680848390_1c3047cb32_o.png\" alt=\"RD Sharma Class 10 Solutions Pair Of Linear Equations In Two Variables\" width=\"142\" height=\"72\"><br>Now plot the points and join them<br>The required region surrounded by these two lines and Y-axis has been shaded as shown.<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1740\/40680849760_c25c7cd62f_o.png\" alt=\"RD Sharma Class 10 Solutions Pair Of Linear Equations In Two Variables\" width=\"349\" height=\"283\"><\/p>\n<p><strong>Question 24.<\/strong><br>Draw the graphs of the following equations on the same graph paper :<br>2x + 3y =12<br>x \u2013 y = 1<br>Find the coordinates of the vertices of the triangle formed by the two straight lines and the y-axis. <strong>(C.B.S.E. 2001)<\/strong><br><strong>Solution:<\/strong><br>2x + 3y = 12<br>\u21d2 2x = 12 \u2013 3y<br>x =&nbsp;<span id=\"MathJax-Element-37-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-307\" class=\"math\"><span id=\"MathJax-Span-308\" class=\"mrow\"><span id=\"MathJax-Span-309\" class=\"mfrac\"><span id=\"MathJax-Span-310\" class=\"mrow\"><span id=\"MathJax-Span-311\" class=\"mn\">12<\/span><span id=\"MathJax-Span-312\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-313\" class=\"mn\">3<\/span><span id=\"MathJax-Span-314\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-315\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1743\/41765944994_648ff8a522_o.png\" alt=\"Pair Of Linear Equations In Two Variables Class 10 RD Sharma\" width=\"149\" height=\"71\"><br>Now plot the points on the graph and join them. Similarly in the equation<br>x \u2013 y = 1 =&gt; x = y + 1<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1746\/41585888395_9fe5d1618e_o.png\" alt=\"Pair Of Linear Equations In Two Variables Class 10 RD Sharma\" width=\"148\" height=\"74\"><br>Now plot the points on the graph and join them<br>The required region surrounded by these two lines and y-axis has been shaded as shown<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/873\/41585889085_10b1979df8_o.png\" alt=\"Pair Of Linear Equations In Two Variables Class 10 RD Sharma\" width=\"290\" height=\"296\"><\/p>\n<p><strong>Question 25.<\/strong><br>Draw the graphs of x \u2013 y + 1 = 0 and 3x + 2y \u2013 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and x-axis and shade the triangular area. Calculate the area bounded by these lines and x-axis.&nbsp;<strong>(C.B.S.E. 2002)<\/strong><br><strong>Solution:<\/strong><br>x \u2013 y + 1 =0, 3x + 2y-12 = 0<br>x \u2013 y + 1 = 0<br>x = y \u2013 1<br>Substituting some different values of y, we get so their corresponding values of x as shown below :<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1723\/40681691980_04c1f3c911_o.png\" alt=\"Pair Of Linear Equations In Two Variables Class 10 RD Sharma\" width=\"145\" height=\"65\"><br>Now plot the points and join them Similarly, in the equation<br>3x + 2y \u2013 12 = 0 =&gt; 3x = 12 \u2013 2y<br>x =&nbsp;<span id=\"MathJax-Element-38-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-316\" class=\"math\"><span id=\"MathJax-Span-317\" class=\"mrow\"><span id=\"MathJax-Span-318\" class=\"mfrac\"><span id=\"MathJax-Span-319\" class=\"mrow\"><span id=\"MathJax-Span-320\" class=\"mn\">12<\/span><span id=\"MathJax-Span-321\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-322\" class=\"mn\">2<\/span><span id=\"MathJax-Span-323\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-324\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1759\/40681692180_d299c8a4da_o.png\" alt=\"RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"155\" height=\"81\"><br>Plot the points on the graph and join them. These two lines intersect each other at (2, 3) and x-axis at (-1, 0) and (4, 0)<br>Area of the triangle ABC =&nbsp;<span id=\"MathJax-Element-39-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-325\" class=\"math\"><span id=\"MathJax-Span-326\" class=\"mrow\"><span id=\"MathJax-Span-327\" class=\"mfrac\"><span id=\"MathJax-Span-328\" class=\"mn\">1<\/span><span id=\"MathJax-Span-329\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x Base x Altitude<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/885\/41585889795_b1b7601979_o.png\" alt=\"RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"330\" height=\"432\"><\/p>\n<p><strong>Question 26.<\/strong><br>Solve graphically the system of linear equations :<br>4x \u2013 3y + 4 = 0<br>4x + 3y \u2013 20 = 0<br>Find the area bounded by these lines and the x-axis. <strong>(C.B.S.E. 2002)<\/strong><br><strong>Solution:<\/strong><br>4x \u2013 3y + 4 = 0<br>=&gt; 4x = 3y \u2013 4<br>=&gt; x =&nbsp;<span id=\"MathJax-Element-40-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-330\" class=\"math\"><span id=\"MathJax-Span-331\" class=\"mrow\"><span id=\"MathJax-Span-332\" class=\"mfrac\"><span id=\"MathJax-Span-333\" class=\"mrow\"><span id=\"MathJax-Span-334\" class=\"mn\">3<\/span><span id=\"MathJax-Span-335\" class=\"mi\">y<\/span><span id=\"MathJax-Span-336\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-337\" class=\"mn\">4<\/span><\/span><span id=\"MathJax-Span-338\" class=\"mn\">4<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/898\/40680855320_7782fe6fdf_o.png\" alt=\"RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"622\" height=\"466\"><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1756\/41765947774_533cfc8819_o.png\" alt=\"RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"355\" height=\"421\"><\/p>\n<p><strong>Question 27.<\/strong><br>Solve the following system of linear equations graphically 3x + y \u2013 11 = 0, x \u2013 y \u2013 1 = 0. Shade the region bounded by these lines and the y-axis. Also, find the area of the region bounded by these lines and the y-axis. <strong>(C.B.S.E. 2002C)<\/strong><br><strong>Solution:<\/strong><br>3x + y \u2013 11=0<br>y = 11 \u2013 3x<br>Substituting some different values of x, we get their corresponding values of y as shown below :<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1730\/40680856390_10531f37db_o.png\" alt=\"RD Sharma Class 10 Solutions Pair Of Linear Equations In Two Variables\" width=\"144\" height=\"68\"><br>Now plot these points on the graph and join them Similarly in the equation<br>x \u2013 y \u2013 1 = 0<br>=&gt; x = y + 1<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/886\/40680855550_62bd5d4612_o.png\" alt=\"RD Sharma Class 10 Solutions Pair Of Linear Equations In Two Variables\" width=\"144\" height=\"68\"><br>Now plot these points on the graph and join them. We see that these two lines intersect each other at point (3,2)<br>x = 3, y = 2<br>Now shade the region enclosed by these two lines and the y-axis<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1752\/40680856290_310b3f17b1_o.png\" alt=\"RD Sharma Class 10 Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"277\" height=\"445\"><br>Area of shaded \u2206ABC<br>=&nbsp;<span id=\"MathJax-Element-41-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-339\" class=\"math\"><span id=\"MathJax-Span-340\" class=\"mrow\"><span id=\"MathJax-Span-341\" class=\"mfrac\"><span id=\"MathJax-Span-342\" class=\"mn\">1<\/span><span id=\"MathJax-Span-343\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x AC x BD<br>=&nbsp;<span id=\"MathJax-Element-42-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-344\" class=\"math\"><span id=\"MathJax-Span-345\" class=\"mrow\"><span id=\"MathJax-Span-346\" class=\"mfrac\"><span id=\"MathJax-Span-347\" class=\"mn\">1<\/span><span id=\"MathJax-Span-348\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x 12 x 3 = 18 sq. units<\/p>\n<p><strong>Question 28.<\/strong><br>Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet the axis of x in each system.<br>(i) 2x + y = 6<br>x \u2013 2y = -2<strong>&nbsp;(C.B.S.E. 1998)<\/strong><br>(ii) 2x \u2013 y = 2<br>4x \u2013 y = 8<strong>&nbsp;(C.B.S.E. 1998)<\/strong><br>(iii) x + 2y = 5<br>2x \u2013 3y = -4&nbsp;<strong>(C.B.S.E. 2005)<\/strong><br>(iv) 2x + 3y = 8<br>x \u2013 2y = -3&nbsp;<strong>(C.B.S.E. 2005)<\/strong><br><strong>Solution:<\/strong><br>(i) 2x + y = 6, x \u2013 2y = -2<br>2x + y = 6<br>y = 6 \u2013 2x<br>Substituting some different values of x, we get their corresponding values of y as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1727\/42488523871_0ac228b030_o.png\" alt=\"Learncbse.In Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"148\" height=\"80\"><br>Now plot the points and join them Similarly in the equation<br>x \u2013 2y = -2<br>=&gt; x = 2y \u2013 2<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1744\/40680856750_d578ed7c90_o.png\" alt=\"Learncbse.In Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"142\" height=\"74\"><br>Now plot the points and join them We see that these two lines intersect each other at (2, 2)<br>x = 2, y = 2<br>Here two lines also meet the x-axis at (3, 0) and (-2, 0) respectively as shown in the figure.<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1733\/40680857880_42c84da227_o.png\" alt=\"Learncbse.In Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"298\" height=\"311\"><br>(ii) 2x \u2013 y = 2, 4x \u2013 y = 8<br>2x \u2013 y = 2<br>=&gt; y = 2x \u2013 2<br>Substituting some different values of x, we get corresponding values of y as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1737\/41765952654_7706cd5580_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"144\" height=\"79\"><br>Now plot the points on the graph and join them Similarly in the equation<br>4x \u2013 y = 8<br>=&gt; y = 4x \u2013 8<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1742\/42488522071_0a45f0f573_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"148\" height=\"72\"><br>Now plot these points and join them We see that these two lines intersect each other at (3, 4)<br>x = 3, y = 4<br>These two lines also meet the x-axis at (1, 0) and (2,0) respectively as shown in the figure<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1756\/42488522451_b203a71a07_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"299\" height=\"349\"><br>(iii) x + 2y = 5, 2x \u2013 3y = -4<br>x + 2y = 5<br>=&gt; x = 5 \u2013 2y<br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1734\/40680859480_d5e07bc8bc_o.png\" alt=\"RD Sharma Class 10 Pdf Ebook Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"160\" height=\"80\"><br>Now plot the points on the graph and join them<br>Similarly in the equation<br>2x \u2013 3y = 4<br>=&gt; 2x = 3y \u2013 4<br>x =&nbsp;<span id=\"MathJax-Element-43-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-349\" class=\"math\"><span id=\"MathJax-Span-350\" class=\"mrow\"><span id=\"MathJax-Span-351\" class=\"mfrac\"><span id=\"MathJax-Span-352\" class=\"mrow\"><span id=\"MathJax-Span-353\" class=\"mn\">3<\/span><span id=\"MathJax-Span-354\" class=\"mi\">y<\/span><span id=\"MathJax-Span-355\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-356\" class=\"mn\">4<\/span><\/span><span id=\"MathJax-Span-357\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1752\/40680859670_f9fa0da41b_o.png\" alt=\"RD Sharma Class 10 Pdf Ebook Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"144\" height=\"81\"><br>Plot these points and join them<br>We see that these two lines intersect each other at (1, 2)<br>x = 1, y = 2<br>and these two lines meet the x-axis at (5, 0) and (-2, 0) respectively as shown in the figure<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1754\/40680860670_d9112fab7f_o.png\" alt=\"RD Sharma Class 10 Pdf Ebook Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"341\" height=\"292\"><br>(iv) 2x + 3y = 8, x \u2013 2y = -3<br>2x + 3y = 8<br>=&gt; 2x = 8 \u2013 3y<br>x =&nbsp;<span id=\"MathJax-Element-44-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-358\" class=\"math\"><span id=\"MathJax-Span-359\" class=\"mrow\"><span id=\"MathJax-Span-360\" class=\"mfrac\"><span id=\"MathJax-Span-361\" class=\"mrow\"><span id=\"MathJax-Span-362\" class=\"mn\">8<\/span><span id=\"MathJax-Span-363\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-364\" class=\"mn\">3<\/span><span id=\"MathJax-Span-365\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-366\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/886\/42488523181_dcb701f50f_o.png\" alt=\"RD Sharma Class 10 Book Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"161\" height=\"78\"><br>Plot these points on the graph and join them Similarly in the equation<br>x \u2013 2y = -3<br>x = 2y \u2013 3<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1736\/42488523281_bc92152f68_o.png\" alt=\"RD Sharma Class 10 Book Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"158\" height=\"72\"><br>Now plot these points and join them We see that these two lines intersect each other at (1, 2)<br>x = 1, y = 2<br>and also these lines meet the x-axis at (4, 0) and (-3, 0) respectively as shown in the figure<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1745\/42488523801_b4cf448fa3_o.png\" alt=\"RD Sharma Class 10 Book Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"346\" height=\"287\"><\/p>\n<p><strong>Question 29.<\/strong><br>Draw the graphs of the following equations 2x \u2013 3y + 6 = 0<br>2x + 3y \u2013 18 = 0<br>y \u2013 2 = 0<br>Find the vertices of the triangle so obtained. Also, find the area of the triangle.<br><strong>Solution:<\/strong><br>2x \u2013 3y + 6 = 0<br>2x = 3y \u2013 6<br>x =&nbsp;<span id=\"MathJax-Element-45-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-367\" class=\"math\"><span id=\"MathJax-Span-368\" class=\"mrow\"><span id=\"MathJax-Span-369\" class=\"mfrac\"><span id=\"MathJax-Span-370\" class=\"mrow\"><span id=\"MathJax-Span-371\" class=\"mn\">3<\/span><span id=\"MathJax-Span-372\" class=\"mi\">y<\/span><span id=\"MathJax-Span-373\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-374\" class=\"mn\">6<\/span><\/span><span id=\"MathJax-Span-375\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1747\/27618947647_39cd3aaf12_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"155\" height=\"79\"><br>Now plot these points on the graph and join them<br>Similarly in the equation 2x + 3y -18 = 0<br>=&gt; 2x = 18 \u2013 3y<br>x =&nbsp;<span id=\"MathJax-Element-46-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-376\" class=\"math\"><span id=\"MathJax-Span-377\" class=\"mrow\"><span id=\"MathJax-Span-378\" class=\"mfrac\"><span id=\"MathJax-Span-379\" class=\"mrow\"><span id=\"MathJax-Span-380\" class=\"mn\">18<\/span><span id=\"MathJax-Span-381\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-382\" class=\"mn\">3<\/span><span id=\"MathJax-Span-383\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-384\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1722\/40681837000_b7b12ddfd2_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"147\" height=\"79\"><br>and in equation y \u2013 2 = 0<br>y = 2<br>Which is parallel to the x-axis on its positive side Now plot the points and join them We see that these lines intersect each other at (3, 4), (6, 2), and (0, 2)<br>Area of the triangle ABC, so formed<br>=&nbsp;<span id=\"MathJax-Element-47-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-385\" class=\"math\"><span id=\"MathJax-Span-386\" class=\"mrow\"><span id=\"MathJax-Span-387\" class=\"mfrac\"><span id=\"MathJax-Span-388\" class=\"mn\">1<\/span><span id=\"MathJax-Span-389\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x base x altitude<br>=&nbsp;<span id=\"MathJax-Element-48-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-390\" class=\"math\"><span id=\"MathJax-Span-391\" class=\"mrow\"><span id=\"MathJax-Span-392\" class=\"mfrac\"><span id=\"MathJax-Span-393\" class=\"mn\">1<\/span><span id=\"MathJax-Span-394\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x BC x AD<br>=&nbsp;<span id=\"MathJax-Element-49-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-395\" class=\"math\"><span id=\"MathJax-Span-396\" class=\"mrow\"><span id=\"MathJax-Span-397\" class=\"mfrac\"><span id=\"MathJax-Span-398\" class=\"mn\">1<\/span><span id=\"MathJax-Span-399\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x 6 x 2<br>= 6 sq. units<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1730\/27618947517_49ab731738_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"358\" height=\"285\"><\/p>\n<p><strong>Question 30.<\/strong><br>Solve the following system of equations graphically:<br>2x \u2013 3y + 6 = 0<br>2x + 3y \u2013 18 = 0<br>Also, find the area of the region bounded by these two lines and the y-axis.<br><strong>Solution:<\/strong><br>2x \u2013 3y + 6 = 0<br>2x = 3y \u2013 6<br>x =&nbsp;<span id=\"MathJax-Element-50-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-400\" class=\"math\"><span id=\"MathJax-Span-401\" class=\"mrow\"><span id=\"MathJax-Span-402\" class=\"mfrac\"><span id=\"MathJax-Span-403\" class=\"mrow\"><span id=\"MathJax-Span-404\" class=\"mn\">3<\/span><span id=\"MathJax-Span-405\" class=\"mi\">y<\/span><span id=\"MathJax-Span-406\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-407\" class=\"mn\">6<\/span><\/span><span id=\"MathJax-Span-408\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/902\/42488524421_6140a514b9_o.png\" alt=\"RD Sharma Class 10 Textbook PDF Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"145\" height=\"77\"><br>Plot these points on the graph and join them Similarly in the equation<br>2x + 3y \u2013 18 = 0<br>=&gt; 2x = 18 \u2013 3y<br>x =&nbsp;<span id=\"MathJax-Element-51-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-409\" class=\"math\"><span id=\"MathJax-Span-410\" class=\"mrow\"><span id=\"MathJax-Span-411\" class=\"mfrac\"><span id=\"MathJax-Span-412\" class=\"mrow\"><span id=\"MathJax-Span-413\" class=\"mn\">18<\/span><span id=\"MathJax-Span-414\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-415\" class=\"mn\">3<\/span><span id=\"MathJax-Span-416\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-417\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/894\/42488523981_fdfc50518f_o.png\" alt=\"RD Sharma Class 10 Textbook PDF Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"151\" height=\"75\"><br>Plot these points on the graph and join them. We see that these two lines intersect each other at (3, 4)<br>x = 3, y = 4<br>These lines formed a triangle ABC with the y-axis<br>Area of \u2206ABC =&nbsp;<span id=\"MathJax-Element-52-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-418\" class=\"math\"><span id=\"MathJax-Span-419\" class=\"mrow\"><span id=\"MathJax-Span-420\" class=\"mfrac\"><span id=\"MathJax-Span-421\" class=\"mn\">1<\/span><span id=\"MathJax-Span-422\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x base x altitude<br>=&nbsp;<span id=\"MathJax-Element-53-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-423\" class=\"math\"><span id=\"MathJax-Span-424\" class=\"mrow\"><span id=\"MathJax-Span-425\" class=\"mfrac\"><span id=\"MathJax-Span-426\" class=\"mn\">1<\/span><span id=\"MathJax-Span-427\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x BC x AD<br>=&nbsp;<span id=\"MathJax-Element-54-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-428\" class=\"math\"><span id=\"MathJax-Span-429\" class=\"mrow\"><span id=\"MathJax-Span-430\" class=\"mfrac\"><span id=\"MathJax-Span-431\" class=\"mn\">1<\/span><span id=\"MathJax-Span-432\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x 4 x 3 = 6 Sq. units<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/898\/42488524381_0fa6ae3809_o.png\" alt=\"RD Sharma Class 10 Textbook PDF Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"351\" height=\"296\"><\/p>\n<p><strong>Question 31.<\/strong><br>Solve the following system of linear equations graphically :<br>4x \u2013 5y \u2013 20 = 0<br>3x + 5y \u2013 15 = 0<br>Determine the vertices of the triangle formed by the lines representing the above equation and the y-axis.&nbsp;<strong>(C.B.S.E. 2004)<\/strong><br><strong>Solution:<\/strong><br>4x \u2013 5y \u2013 20 = 0<br>=&gt; 4x = 5y + 20<br>x =&nbsp;<span id=\"MathJax-Element-55-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-433\" class=\"math\"><span id=\"MathJax-Span-434\" class=\"mrow\"><span id=\"MathJax-Span-435\" class=\"mfrac\"><span id=\"MathJax-Span-436\" class=\"mrow\"><span id=\"MathJax-Span-437\" class=\"mn\">5<\/span><span id=\"MathJax-Span-438\" class=\"mi\">y<\/span><span id=\"MathJax-Span-439\" class=\"mo\">+<\/span><span id=\"MathJax-Span-440\" class=\"mn\">20<\/span><\/span><span id=\"MathJax-Span-441\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1746\/42488525231_a6acc3ed5a_o.png\" alt=\"10th Maths Solution Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"153\" height=\"76\"><br>Plot these points on the graph and join them Similarly in the equation<br>3x + 5y \u2013 15 = 0<br>=&gt; 3x = 15 \u2013 5y<br>x =&nbsp;<span id=\"MathJax-Element-56-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-442\" class=\"math\"><span id=\"MathJax-Span-443\" class=\"mrow\"><span id=\"MathJax-Span-444\" class=\"mfrac\"><span id=\"MathJax-Span-445\" class=\"mrow\"><span id=\"MathJax-Span-446\" class=\"mn\">15<\/span><span id=\"MathJax-Span-447\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-448\" class=\"mn\">5<\/span><span id=\"MathJax-Span-449\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-450\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1733\/42488524631_c378638933_o.png\" alt=\"10th Maths Solution Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"157\" height=\"72\"><br>Now plot these points and join them We see that these two lines intersect each other at (5, 0)<br>x = 5, y = 0<br>These two lines form an \u2206ABC with a y-axis whose vertices are A (5, 0), B (0, 3), C (0, -4)<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1734\/42437087852_aafe15fe38_o.png\" alt=\"10th Maths Solution Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"353\" height=\"462\"><\/p>\n<p><strong>Question 32.<\/strong><br>Draw the graphs of the equations 5x \u2013 y = 5 and 3x \u2013 y = 3. Determine the coordinates of the vertices of the triangle formed by these lines and the y-axis calculate the area of the triangle so formed.<br><strong>Solution:<\/strong><br>5x \u2013 y = 5<br>=&gt; y = 5x \u2013 5<br>Substituting some different values of x, we get their corresponding values of y as shown below:<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1734\/27618069297_f6ee8dc3dd_o.png\" alt=\"RD Sharma Class 10 Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"153\" height=\"76\"><br>Plot these points on the graph and join them. Similarly in the equation<br>3x \u2013 y = 3<br>=&gt; y = 3x \u2013 3<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1737\/42437087942_c9a01d9c19_o.png\" alt=\"RD Sharma Class 10 Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"151\" height=\"74\"><br>Now plot these points and join them We see that these two lines intersect each other at (1, 0)<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1740\/42437088382_b463645851_o.png\" alt=\"RD Sharma Class 10 Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"342\" height=\"580\"><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1721\/27618069207_3735c61375_o.png\" alt=\"RD Sharma 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"345\" height=\"347\"><\/p>\n<p><strong>Question 33.<\/strong><br>Form the pair of linear equations in the following problems, and find their solution graphically.<br>(i) 10 students of class X took part in the Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.<br>(ii) 5 pencils and 7 pens together cost Rs. 50, whereas 7 pencils and 5 pens together cost Rs. 46. Find the cost of one pencil and a pen.<br>(iii) Champa went to a \u2018sale\u2019 to purchase some pants and skirts. When her friends asked her how many of each she had bought, she answered, \u201cThe number of skirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased.\u201d Help her friends to find how many pants and skirts Champa bought. <strong>[NCERT]<\/strong><br><strong>Solution:<\/strong><br>Let number of boys = x<br>and number of girls = y<br>According to the given conditions<br>x + y = 10<br>y \u2013 x = 4<br>Now, x + y = 10<br>=&gt; x = 10 \u2013 y<br>Substituting some different values of y, we get their corresponding values of x as given below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1754\/27618072147_a25b1c0711_o.png\" alt=\"RD Sharma 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"145\" height=\"76\"><br>Plot the points on the graph and join them Similarly in the equation<br>y \u2013 x = 4<br>=&gt; y = 4 + x<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1747\/27618069557_278247f1f9_o.png\" alt=\"RD Sharma Class 10 Maths Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"156\" height=\"76\"><br>Now plot the points and join them we see that these two lines intersect each other at (3, 7)<br>x = 3, y = 7<br>Number of boys = 3<br>and number of girls = 7<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/877\/27618070257_7eb6e7ea5c_o.png\" alt=\"RD Sharma Class 10 Maths Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"310\" height=\"356\"><br>(ii) Let cost of 1 pencil = Rs. x<br>and cost of 1 pen = Rs. y<br>According to the given conditions,<br>5x + 7y = 50<br>2x + 5y = 46<br>5x + 7y = 50<br>5x = 50 \u2013 7y<br>x =&nbsp;<span id=\"MathJax-Element-57-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-451\" class=\"math\"><span id=\"MathJax-Span-452\" class=\"mrow\"><span id=\"MathJax-Span-453\" class=\"mfrac\"><span id=\"MathJax-Span-454\" class=\"mrow\"><span id=\"MathJax-Span-455\" class=\"mn\">50<\/span><span id=\"MathJax-Span-456\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-457\" class=\"mn\">7<\/span><span id=\"MathJax-Span-458\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-459\" class=\"mn\">5<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get the corresponding values of x as given below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1737\/27618070517_368883db75_o.png\" alt=\"RD Sharma Class 10 Maths Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"160\" height=\"80\"><br>Plot these points and join them Similarly in the equation<br>7x + 5y = 46<br>=&gt; 7x = 46 \u2013 5y<br>=&gt; x =&nbsp;<span id=\"MathJax-Element-58-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-460\" class=\"math\"><span id=\"MathJax-Span-461\" class=\"mrow\"><span id=\"MathJax-Span-462\" class=\"mfrac\"><span id=\"MathJax-Span-463\" class=\"mrow\"><span id=\"MathJax-Span-464\" class=\"mn\">46<\/span><span id=\"MathJax-Span-465\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-466\" class=\"mn\">5<\/span><span id=\"MathJax-Span-467\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-468\" class=\"mn\">7<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/882\/27618070897_7e43f62d9c_o.png\" alt=\"Class 10 RD Sharma Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"157\" height=\"80\"><br>Now plot the points on the graph and join them. We see that these two lines intersect each other at (3, 5)<br>x = 3, y = 5<br>or cost of pencil = Rs. 3<br>and cost of a pen = Rs. 5<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/878\/27618072027_b6811d47ba_o.png\" alt=\"Class 10 RD Sharma Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"478\" height=\"494\"><br>(iii) Let number of skirts = x<br>and number of pants = y<br>According to the given condition,<br>x = 2y \u2013 2 and x = 4y \u2013 4<br>2y \u2013 2 = 4y \u2013 4<br>4y \u2013 2y = -2 + 4<br>2y = 2<br>y = 1<br>and x = 2y \u2013 2 = 2 x 1 \u2013 2 = 2 \u2013 2 = 0<br>Number of skirts = 0<br>and number of pants = 1<\/p>\n<p><strong>Question 34.<\/strong><br>Solve the following system of equations graphically shade the region between the lines and the y -axis<br>(i) 3x \u2013 4y = 7<br>5x + 2y = 3&nbsp;<strong>(C.B.S.E. 2006C)<\/strong><br>(ii) 4x \u2013 y = 4<br>3x + 2y = 14&nbsp;<strong>(C.B.S.E. 2006C)<\/strong><br><strong>Solution:<\/strong><br>(i) 3x \u2013 4y = 7<br>3x = 7 + 4y<br>x =&nbsp;<span id=\"MathJax-Element-59-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-469\" class=\"math\"><span id=\"MathJax-Span-470\" class=\"mrow\"><span id=\"MathJax-Span-471\" class=\"mfrac\"><span id=\"MathJax-Span-472\" class=\"mrow\"><span id=\"MathJax-Span-473\" class=\"mn\">7<\/span><span id=\"MathJax-Span-474\" class=\"mo\">+<\/span><span id=\"MathJax-Span-475\" class=\"mn\">4<\/span><span id=\"MathJax-Span-476\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-477\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1760\/42488530031_95645fc3ab_o.png\" alt=\"Class 10 RD Sharma Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"163\" height=\"76\"><br>Plot these points on the graph and join them. Similarly in the equation<br>5x + 2y = 3<br>=&gt; 5x = 3 \u2013 2y<br>x =&nbsp;<span id=\"MathJax-Element-60-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-478\" class=\"math\"><span id=\"MathJax-Span-479\" class=\"mrow\"><span id=\"MathJax-Span-480\" class=\"mfrac\"><span id=\"MathJax-Span-481\" class=\"mrow\"><span id=\"MathJax-Span-482\" class=\"mn\">3<\/span><span id=\"MathJax-Span-483\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-484\" class=\"mn\">2<\/span><span id=\"MathJax-Span-485\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-486\" class=\"mn\">5<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/890\/27618072507_091db601af_o.png\" alt=\"Answers Of RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"160\" height=\"73\"><br>Plot these points and join them We see that the lines intersect each other at (1, -1)<br>x = 1, y = -1<br>Now the region between these lines and the y-axis has been shaded as shown<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1726\/42488528991_7912cf44a6_o.png\" alt=\"Answers Of RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"357\" height=\"417\"><br>(ii) 4x \u2013 y = 4<br>3x + 2y = 14<br>4x \u2013 y = 4<br>y = 4x \u2013 4<br>Substituting some different values of x, we get their corresponding values of y as given below<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/890\/42488529141_ebfdaeefc2_o.png\" alt=\"Answers Of RD Sharma Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"149\" height=\"73\"><br>Plot these points and join them Similarly in equation<br>3x + 2y = 14<br>3x = 14 \u2013 2y<br>x =&nbsp;<span id=\"MathJax-Element-61-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-487\" class=\"math\"><span id=\"MathJax-Span-488\" class=\"mrow\"><span id=\"MathJax-Span-489\" class=\"mfrac\"><span id=\"MathJax-Span-490\" class=\"mrow\"><span id=\"MathJax-Span-491\" class=\"mn\">14<\/span><span id=\"MathJax-Span-492\" class=\"mo\">\u2013<\/span><span id=\"MathJax-Span-493\" class=\"mn\">2<\/span><span id=\"MathJax-Span-494\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-495\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1728\/42488529221_3596c45d73_o.png\" alt=\"RD Sharma Maths Book For Class 10 Solution Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"154\" height=\"78\"><br>Now plot these points and join them<br>We see that these lines intersect each other at (2, 4)<br>x = 2, y = 4<br>The region between these two lines and the y-axis has been shaded as shown<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1745\/42488529611_d3e9df7466_o.png\" alt=\"RD Sharma Maths Book For Class 10 Solution Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"308\" height=\"437\"><\/p>\n<p><strong>Question 35.<\/strong><br>Represent the following pair of equations graphically and write the coordinates of points where the lines intersects y-axis<br>x + 3y = 6<br>2x \u2013 3y = 12&nbsp;<strong>(C.B.S.E. 2008)<\/strong><br><strong>Solution:<\/strong><br>x + 3y = 6<br>x = 6 \u2013 3y<br>Substituting some different values of y, we get their corresponding values of x as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1723\/27618075707_8c2d315709_o.png\" alt=\"RD Sharma Maths Book For Class 10 Solution Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"159\" height=\"72\"><br>Now plot these points on the graph and join them<br>Similarly in the equation<br>2x \u2013 3y = 12 =&gt; 2x = 12 + 3y<br>x =&nbsp;<span id=\"MathJax-Element-62-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-496\" class=\"math\"><span id=\"MathJax-Span-497\" class=\"mrow\"><span id=\"MathJax-Span-498\" class=\"mfrac\"><span id=\"MathJax-Span-499\" class=\"mrow\"><span id=\"MathJax-Span-500\" class=\"mn\">12<\/span><span id=\"MathJax-Span-501\" class=\"mo\">+<\/span><span id=\"MathJax-Span-502\" class=\"mn\">3<\/span><span id=\"MathJax-Span-503\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-504\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/889\/27618074797_e14fbe297c_o.png\" alt=\"RD Sharma Maths Book For Class 10 Solution Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"155\" height=\"74\"><br>Now plot their points and join them We see that these two lines meet the y-axis at (0, 2) and (0, -4)<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1752\/27618075477_0d5d44059f_o.png\" alt=\"10th Maths Solution Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"333\" height=\"341\"><\/p>\n<p><strong>Question 36.<\/strong><br>Given the linear equation 2x + 3y \u2013 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is<br>(i) intersecting lines<br>(ii) Parallel lines<br>(iii) coincident lines&nbsp;<strong>[NCERT]<\/strong><br><strong>Solution:<\/strong><br>Given a linear equation 2x + 3y \u2013 8 = 0<br>(i) When the lines are intersecting, then<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1744\/27618076047_5d042be548_o.png\" alt=\"10th Maths Solution Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"303\" height=\"373\"><\/p>\n<p><strong>Question 37.<\/strong><br>Determine graphically the co-ordinates of the vertices of a triangle, the equations of whose sides are :<br>(i) y = x, y = 2x and y + x = 6&nbsp;<strong>(C.B.S.E. 2000)<\/strong><br>(ii) y = x, 3y = x, x + y = 8&nbsp;<strong>(C.B.S.E. 2000)<\/strong><br><strong>Solution:<\/strong><br>(i) y = x<br>Substituting some different values of x, we get their corresponding values of y as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/902\/42488532691_fe1efc8652_o.png\" alt=\"10th Maths Solution Book Pdf Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"155\" height=\"76\"><br>Now plot the points on the graph and join them. Similarly in the equation y = 2x<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/881\/27618076407_ace54b18cc_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"154\" height=\"75\"><br>and y + x = 6 =&gt; x = 6 \u2013 y<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1739\/42488531091_cba14b8ffd_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"144\" height=\"71\"><br>Now plot the points on the graph and join them. We see that these lines intersect each other at (0, 0), (3, 3), and (2, 4)<br>Vertices of the triangle so formed by these lines are (0, 0), (3, 3), and (2, 4)<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1741\/42488531601_f667b3170d_o.png\" alt=\"RD Sharma Class 10 Pdf Free Download Full Book Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"319\" height=\"280\"><br>(ii) y = x, 3y = x, x + y = 8<br>y = x<br>Substituting some different values of x, we get corresponding values of y as shown below<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/898\/27618077657_319e233568_o.png\" alt=\"Class 10 RD Sharma Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"150\" height=\"77\"><br>Plot these points on the graph and join them Similarly in the equation 3y = x<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/874\/42488531751_42c5425f3a_o.png\" alt=\"Class 10 RD Sharma Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"163\" height=\"71\"><br>and x + y = 8 =&gt; x = 8 \u2013 y<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/889\/27618077837_fa0397357a_o.png\" alt=\"Class 10 RD Sharma Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"144\" height=\"77\"><br>Now plot the points and join them. We see that these lines intersect each other at (0,0), (4, 4), (6, 2)<br>The vertices of the triangle so formed are (0, 0), (4, 4), and (6, 2)<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/890\/27618079017_b7cb86b916_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"339\" height=\"328\"><\/p>\n<p><strong>Question 38.<\/strong><br>Graphically, solve the following pair of equations:<br>2x + y = 6<br>2x \u2013 y + 2 = 0<br>Find the ratio of the areas of the two triangles formed by the lines representing these equations with the x-axis and the lines with the y-axis.&nbsp;<strong>[NCERT Exemplar]<\/strong><br><strong>Solution:<\/strong><br>Given equations are 2x + y \u2013 6 and 2x \u2013 y + 2 = 0<br>Table for equation 2x + y = 6<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1756\/42488533351_c64f664d57_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"220\" height=\"113\"><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1732\/42488533001_48b67e215b_o.png\" alt=\"RD Sharma Maths Class 10 Solutions Pdf Free Download Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"331\" height=\"462\"><br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1737\/27618079967_d0ee884d4d_o.png\" alt=\"Learncbse.In Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"346\" height=\"306\"><br>Hence, the pair of equations intersect graphically at point E (1, 4), i.e., x = 1 and y = 4<\/p>\n<p><strong>Question 39.<\/strong><br>Determine, graphically, the vertices of the triangles formed by the lines y = x, 3y = x, x + y = 8.&nbsp;<strong>[NCERT Exemplar]<\/strong><br><strong>Solution:<\/strong><br>Given linear equations are y = x \u2026\u2026.(i)<br>3y = x \u2026\u2026\u2026(ii)<br>and x + y = 8 \u2026\u2026.(iii)<br>For equation y = x,<br>If x = 1, then y = 1<br>If x = 0, then y = 0<br>If x = 2, then y = 2<br>Table for line y = x,<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1723\/42488533971_313cfe8f10_o.png\" alt=\"Learncbse.In Class 10 Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"290\" height=\"113\"><br>For equation x = 3y<br>If x = 0, then y = 0,<br>if x = 3, then y = 1<br>and if x = 6, then y = 2<br>Table for line x = 3y,<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/882\/27618080387_bc35d3818d_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"283\" height=\"118\"><br>For equation,<br>If x = 0, then y = 8<br>if x = 8, then y = 0<br>and if x = 4, then y = 4<br>Table for line x + y = 8,<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/901\/27618080677_fcd898a804_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"286\" height=\"112\"><br>Plotting points A (1, 1) and B (2,2), we get the straight line AB. Plotting the points C (3, 1) and D (6, 2), we get the straight line CD. Plotting the points P (0, 8), Q (4, 4), and R (8, 0), we get the straight line PQR. We see that lines AB and CD intersecting the line PR on Q and D, respectively.<br>So, \u2206OQD is formed by these lines. Hence, the vertices of the \u2206OQD formed by the given lines are O (0, 0), Q (4, 4), and D (6, 2).<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1739\/27618081067_553b3931c8_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"362\" height=\"356\"><\/p>\n<p><strong>Question 40.<\/strong><br>Draw the graph of the equations x = 3, x = 5 and 2x \u2013 y \u2013 4 = 0. Also, find the area of the quadrilateral formed by the lines and the x-axis.<strong>&nbsp;|NCERT Exemplar]<\/strong><br><strong>Solution:<\/strong><br>Given equation of lines 2x \u2013 y \u2013 4 = 0, x = 3 and x = 5<br>Table for line 2x \u2013 y \u2013 4 = 0,<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1737\/27618081837_438d0bbe13_o.png\" alt=\"RD Sharma Class 10 Solutions Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"240\" height=\"105\"><br>Draw the points P (0, -4) and Q (2,0) and join these points and form a line PQ also draw the lines x = 3 and x = 5.<br>Area of quadrilateral ABCD =&nbsp;<span id=\"MathJax-Element-63-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-505\" class=\"math\"><span id=\"MathJax-Span-506\" class=\"mrow\"><span id=\"MathJax-Span-507\" class=\"mfrac\"><span id=\"MathJax-Span-508\" class=\"mn\">1<\/span><span id=\"MathJax-Span-509\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x distance between parallel lines (AB) x (AD + BC) [since, quadrilateral ABCD is a trapezium]<br>=&nbsp;<span id=\"MathJax-Element-64-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-510\" class=\"math\"><span id=\"MathJax-Span-511\" class=\"mrow\"><span id=\"MathJax-Span-512\" class=\"mfrac\"><span id=\"MathJax-Span-513\" class=\"mn\">1<\/span><span id=\"MathJax-Span-514\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x 2 x (6 + 2) [\u2235 AB = OB \u2013 OA = 5 \u2013 3 = 2, AD = 2 and BC = 6]<br>= 8 sq. units<br><img class=\"alignnone\" src=\"https:\/\/farm1.staticflickr.com\/898\/42488534441_2b6381958a_o.png\" alt=\"RD Sharma Class 10 Solution Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"320\" height=\"314\"><br>Hence, the required area of the quadrilateral formed by the lines and the x-axis is 8 sq. units.<\/p>\n<p><strong>Question 41.<\/strong><br>Draw the graphs of the lines x = -2, and y = 3. Write the vertices of the figure formed by these lines, the x-axis and the y-axis. Also, find the area of the figure.&nbsp;<strong>[NCERT Exemplar]<\/strong><br><strong>Solution:<\/strong><br>We know that the graph of x = -2 is a line parallel to the y-axis at a distance of 2 units to the left of it. So, the line l is the graph of x = -2<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1727\/27618082197_23d0dd4f52_o.png\" alt=\"RD Sharma Class 10 Solution Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"283\" height=\"284\"><br>The graph of y = 3 is a line parallel to the x-axis at a distance of 3 units above it.<br>So, the line m is the graph of y = 3<br>The figure enclosed by the line x = -2, y = 3, the x-axis, and the y-axis is OABC, which is a rectangle.<br>A is a point on the y-axis at a distance of 3 units above the x-axis. So, the coordinates of A are (0, 3).<br>C is a point on the x-axis at a distance of 2 units to the left of the y-axis. So, the coordinates of C are (-2, 0).<br>B is the solution of the pair of equations x = -2 and y = 3. So, the coordinates of B are (-2, 3).<br>So, the vertices of the rectangle OABC are O (0, 0), A (0, 3), B (-2, 3), C (-2, 0).<br>The length and breadth of this rectangle are 2 units and 3 units, respectively.<br>As the area of a rectangle = length x breadth, the area of rectangle OABC = 2 x 3 = 6 sq. units.<\/p>\n<p><strong>Question 42.<\/strong><br>Draw the graphs of the pair of linear equations x \u2013 y + 2 = 0 and 4x \u2013 y \u2013 4 = 0. Calculate the area of the triangle formed by the lines so drawn and the x-axis.&nbsp;<strong>[NCERT Exemplar]<\/strong><br><strong>Solution:<\/strong><br>For drawing the graphs of the given equations, we find two solutions of each of the equations, which are given in the table.<br>Plot the points A (0,2), B (-2,0), P (0, -4), and Q (1,0) on the graph paper, and join the points to form the lines AB and PQ as shown in the figure.<br><img class=\"alignnone\" src=\"https:\/\/farm2.staticflickr.com\/1752\/42488535311_d127ddf482_o.png\" alt=\"RD Sharma Class 10 Solution Chapter 3 Pair Of Linear Equations In Two Variables\" width=\"271\" height=\"273\"><br>We observe that there is a point R (2,4) common to both the lines AB and PQ. The triangle formed by these lines and the x-axis is BQR.<br>The vertices of this triangle are B (-2, 0), Q (1, 0), and R (2, 4).<br>We know that;<br>Area of triangle =&nbsp;<span id=\"MathJax-Element-65-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-515\" class=\"math\"><span id=\"MathJax-Span-516\" class=\"mrow\"><span id=\"MathJax-Span-517\" class=\"mfrac\"><span id=\"MathJax-Span-518\" class=\"mn\">1<\/span><span id=\"MathJax-Span-519\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span>&nbsp;x Base x Altitude<br>Here, Base = BQ = BO + OQ = 2 + 1 = 3 units<br>Altitude = RM = Ordinate of R = 4 units.<br>So, area of ABQR =&nbsp;<span id=\"MathJax-Element-66-Frame\" class=\"MathJax\" tabindex=\"0\"><span id=\"MathJax-Span-520\" class=\"math\"><span id=\"MathJax-Span-521\" class=\"mrow\"><span id=\"MathJax-Span-522\" class=\"mfrac\"><span id=\"MathJax-Span-523\" class=\"mn\">1<\/span><span id=\"MathJax-Span-524\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span> x 3 x 4 = 6 sq. units<\/p>\n<p>We have provided complete details of RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2. If you have any queries related to <a href=\"https:\/\/www.cbse.gov.in\/\" target=\"_blank\" rel=\"noopener\"><strong>CBSE<\/strong><\/a>&nbsp;Class 10, feel free to ask us in the comment section below.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"faqs-on-rd-sharma-class-10-solutions-chapter-3-exercise-32\"><\/span>FAQs on RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div id=\"rank-math-faq\" class=\"rank-math-block\">\n<div class=\"rank-math-list \">\n<div id=\"faq-question-1631107843793\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"where-can-i-download-rd-sharma-class-10-solutions-chapter-3-exercise-32-free-pdf\"><\/span>Where can I download RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 free PDF?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can download RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 free PDF from the above article.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1631108063850\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"is-it-required-to-practice-all-of-the-questions-in-rd-sharma-class-10-solutions-chapter-3-exercise-32\"><\/span>Is it required to practice all of the questions in RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>Yes, all of the questions in RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 must be learned. These questions may appear on both board exams and class tests. Students will be prepared for their board exams if they learn these questions.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1631108599540\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"what-are-the-benefits-of-using-rd-sharma-class-10-solutions-chapter-3-exercise-32\"><\/span>What are the benefits of using RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>1. Correct answers according to the last CBSE guidelines and syllabus.<br \/>2. The RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 is written in simple language to assist students in their board examination, &amp; competitive examination preparation.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2:&nbsp;This exercise evaluates your understanding of how to generate graphs of linear equations while solving systems of simultaneous linear equations in two variables. Experts at Kopykitab provide all of the RD Sharma Class 10 Solutions by connecting students at various levels. For any reference relating to the &#8230; <a title=\"RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 (Updated for 2023-24)\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-class-10-solutions-chapter-3-exercise-3-2\/\" aria-label=\"More on RD Sharma Class 10 Solutions Chapter 3 Exercise 3.2 (Updated for 2023-24)\">Read more<\/a><\/p>\n","protected":false},"author":238,"featured_media":125553,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":""},"categories":[73411,2985,73410],"tags":[3243,9206,73520,4388],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/125512"}],"collection":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/users\/238"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/comments?post=125512"}],"version-history":[{"count":4,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/125512\/revisions"}],"predecessor-version":[{"id":357704,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/125512\/revisions\/357704"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media\/125553"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=125512"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=125512"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=125512"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}