{"id":724974,"date":"2024-07-09T11:21:29","date_gmt":"2024-07-09T05:51:29","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=724974"},"modified":"2024-07-09T11:21:29","modified_gmt":"2024-07-09T05:51:29","slug":"class-10-maths-chapter-3-pair-of-linear-equations-in-two-variables-worksheet","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/worksheets-class-10-maths\/pair-of-linear-equations-in-two-variables\/","title":{"rendered":"Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Worksheet"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 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\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" 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: #999;color:#999\" 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\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/worksheets-class-10-maths\/pair-of-linear-equations-in-two-variables\/#Class_10_Maths_Worksheet_for_Pair_of_Linear_Equations_in_Two_Variables\" title=\"Class 10 Maths Worksheet for Pair of Linear Equations in Two Variables\">Class 10 Maths Worksheet for Pair of Linear Equations in Two Variables<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/worksheets-class-10-maths\/pair-of-linear-equations-in-two-variables\/#Class_10_Maths_Worksheet_for_Pair_of_Linear_Equations_in_Two_Variables_-_Download_PDF\" title=\"Class 10 Maths Worksheet for Pair of Linear Equations in Two Variables &#8211; Download PDF\">Class 10 Maths Worksheet for Pair of Linear Equations in Two Variables &#8211; Download PDF<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/worksheets-class-10-maths\/pair-of-linear-equations-in-two-variables\/#Class_10_Math_Chapter_3_Worksheet\" title=\"Class 10 Math Chapter 3 Worksheet\">Class 10 Math Chapter 3 Worksheet<\/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:\/\/infinitylearn.com\/surge\/worksheets-class-10-maths\/pair-of-linear-equations-in-two-variables\/#Worksheet_1_Basics_of_Linear_Equations\" title=\"Worksheet 1: Basics of Linear Equations\">Worksheet 1: Basics of Linear Equations<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/worksheets-class-10-maths\/pair-of-linear-equations-in-two-variables\/#Worksheet_2_Graphical_Method\" title=\"Worksheet 2: Graphical Method\">Worksheet 2: Graphical Method<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/worksheets-class-10-maths\/pair-of-linear-equations-in-two-variables\/#Worksheet_3_Elimination_Method\" title=\"Worksheet 3: Elimination Method\">Worksheet 3: Elimination Method<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/worksheets-class-10-maths\/pair-of-linear-equations-in-two-variables\/#Worksheet_4_Word_Problems\" title=\"Worksheet 4: Word Problems\">Worksheet 4: Word Problems<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<p>Class 10 Maths Worksheets for Chapter 3 (Pair of Linear Equations in Two Variables) are available here online with answers. These worksheets will help students score good marks in their board exams. The questions are made according to the latest<strong><a href=\"https:\/\/infinitylearn.com\/surge\/cbse\/cbse-syllabus\/\"> CBSE syllabus<\/a> <\/strong>(2024-2025) and NCERT textbook. Our subject experts have prepared these worksheets following the latest exam pattern. Practicing these questions will help students develop problem-solving skills. To get chapter-wise worksheets, click here. You can also download the PDF for extra practice questions.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Class_10_Maths_Worksheet_for_Pair_of_Linear_Equations_in_Two_Variables\"><\/span>Class 10 Maths Worksheet for Pair of Linear Equations in Two Variables<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The CBSE board has released the Class 10 Maths exam datasheet. It&#8217;s time for students to review the chapters for the board exam. Students should solve these questions and choose the correct answers. They can check their answers with the provided solutions. Get important questions for Class 10 Maths.<\/p>\n<p>Click here to download the PDF of additional worksheets for practice on Pair of Linear Equations in Two Variables, Chapter of Class 10 Maths, along with the answer key:<\/p>\n<p style=\"text-align: center;\"><strong><a href=\"https:\/\/infinitylearn.com\/surge\/worksheets-class-10-maths\/real-numbers\/\" target=\"_blank\" rel=\"noopener\">Real Numbers Worksheet Class 10<\/a>  |  <\/strong><strong><a href=\"https:\/\/infinitylearn.com\/surge\/worksheets-class-10-maths\/polynomials\/\" target=\"_blank\" rel=\"noopener\">Class 10 Math Polynomials Worksheets <\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Class_10_Maths_Worksheet_for_Pair_of_Linear_Equations_in_Two_Variables_-_Download_PDF\"><\/span>Class 10 Maths Worksheet for Pair of Linear Equations in Two Variables &#8211; Download PDF<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\t\t<div class=\"download-pdf-container\">\n\t\t\t<span class=\"title\">Do you need help with your Homework? 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href=\"\/surge\/?pdfemb-serveurl=https%3A%2F%2Finfinitylearn.com%2Fsurge%2Fwp-content%2Fuploads%2Fsecurepdfs%2F2024%2F07%2FLINEAR-EQUATIONS-IN-TWO-VARIABLES-1-CL-X-may-19.pdf\" class=\"pdfemb-viewer\" style=\"\" data-width=\"max\" data-height=\"max\" data-mobile-width=\"0\"  data-scrollbar=\"both\" data-download=\"off\" data-tracking=\"on\" data-newwindow=\"on\" data-pagetextbox=\"off\" data-scrolltotop=\"on\" data-startzoom=\"100\" data-startfpzoom=\"100\" data-download-nonce=\"9a12f864ee\" data-disablerightclick=\"on\" data-toolbar=\"top\" data-toolbar-fixed=\"on\">Class 10 Maths Worksheet for Pair of Linear Equations in Two Variables<br\/><\/a>\n<h2><span class=\"ez-toc-section\" id=\"Class_10_Math_Chapter_3_Worksheet\"><\/span>Class 10 Math Chapter 3 Worksheet<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"Worksheet_1_Basics_of_Linear_Equations\"><\/span>Worksheet 1: Basics of Linear Equations<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Instructions:<\/strong> Solve the following pairs of linear equations by substitution method.<\/p>\n<ol>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + y = 7<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">7<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x &#8211; y = 3<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span><\/li>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + 3y = 12<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x &#8211; y = 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span><\/li>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>y<\/mi><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + 2y = 8<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + y = 7<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">7<\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n<p><strong>Solutions:<\/strong><\/p>\n<ol>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + y = 7<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">7<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x &#8211; y = 3<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span><\/p>\n<ul>\n<li>Add both equations: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mn>10<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x = 10 \\Rightarrow x = 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span><\/li>\n<li>Substitute <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span> in <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + y = 7<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">7<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>5<\/mn><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>7<\/mn><mo>\u21d2<\/mo><mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5 + y = 7 \\Rightarrow y = 2<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">7<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/li>\n<li>Solution: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 5, y = 2<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + 3y = 12<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x &#8211; y = 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span><\/p>\n<ul>\n<li>Solve <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x &#8211; y = 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span> for <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = 3x &#8211; 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span><\/li>\n<li>Substitute <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = 3x &#8211; 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span> in <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + 3y = 12<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>12<\/mn><mo>\u21d2<\/mo><mn>11<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><mo>=<\/mo><mn>12<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + 3(3x &#8211; 5) = 12 \\Rightarrow 11x &#8211; 15 = 12 \\Rightarrow x = 3<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord\">11<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">15<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span><\/li>\n<li>Substitute <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 3<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span> in <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = 3x &#8211; 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>9<\/mn><mo>\u2212<\/mo><mn>5<\/mn><mo>\u21d2<\/mo><mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = 9 &#8211; 5 \\Rightarrow y = 4<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">9<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span><\/li>\n<li>Solution: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 3, y = 4<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>y<\/mi><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + 2y = 8<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + y = 7<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">7<\/span><\/span><\/span><\/span><\/p>\n<ul>\n<li>Solve <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>y<\/mi><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + 2y = 8<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><\/span><\/span><\/span> for <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>8<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x = 8 &#8211; 2y<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span><\/li>\n<li>Substitute <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>8<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x = 8 &#8211; 2y<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span> in <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + y = 7<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">7<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>8<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>7<\/mn><mo>\u21d2<\/mo><mn>16<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>y<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>7<\/mn><mo>\u21d2<\/mo><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2(8 &#8211; 2y) + y = 7 \\Rightarrow 16 &#8211; 4y + y = 7 \\Rightarrow y = 3<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">8<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">7<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord\">16<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">7<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span><\/li>\n<li>Substitute <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = 3<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span> in <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>8<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x = 8 &#8211; 2y<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>8<\/mn><mo>\u2212<\/mo><mn>6<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 8 &#8211; 6 \\Rightarrow x = 2<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/li>\n<li>Solution: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 2, y = 3<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3><span class=\"ez-toc-section\" id=\"Worksheet_2_Graphical_Method\"><\/span>Worksheet 2: Graphical Method<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Instructions:<\/strong> Solve the following pairs of linear equations by graphical method. Draw the graphs on a graph paper.<\/p>\n<ol>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + y = 4<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x &#8211; y = 2<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/li>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + y = 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x &#8211; y = 1<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n<p><strong>Solutions:<\/strong><\/p>\n<ol>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + y = 4<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x &#8211; y = 2<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/p>\n<ul>\n<li>Equation 1: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + y = 4<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span> can be rewritten as <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>4<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y = 4 &#8211; x<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/li>\n<li>Equation 2: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x &#8211; y = 2<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span> can be rewritten as <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = x &#8211; 2<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/li>\n<li>Plotting these equations on graph paper, the lines intersect at <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(3, 1)<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/li>\n<li>Solution: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 3, y = 1<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + y = 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x &#8211; y = 1<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/p>\n<ul>\n<li>Equation 1: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + y = 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span> can be rewritten as <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y = 5 &#8211; 2x<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/li>\n<li>Equation 2: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x &#8211; y = 1<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span> can be rewritten as <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = x &#8211; 1<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/li>\n<li>Plotting these equations on graph paper, the lines intersect at <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(2, 1)<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/li>\n<li>Solution: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 2, y = 1<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3><span class=\"ez-toc-section\" id=\"Worksheet_3_Elimination_Method\"><\/span>Worksheet 3: Elimination Method<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Instructions:<\/strong> Solve the following pairs of linear equations by elimination method.<\/p>\n<ol>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mi>y<\/mi><mo>=<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x + 4y = 10<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x &#8211; y = 3<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span><\/li>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + 3y = 6<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x &#8211; 3y = 12<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><\/span><\/span><\/span><\/li>\n<\/ol>\n<p><strong>Solutions:<\/strong><\/p>\n<ol>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mi>y<\/mi><mo>=<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x + 4y = 10<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x &#8211; y = 3<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span><\/p>\n<ul>\n<li>Multiply second equation by 4: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>y<\/mi><mo>=<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">8x &#8211; 4y = 12<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">8<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><\/span><\/span><\/span><\/li>\n<li>Add both equations: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mi>y<\/mi><mo>+<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mi>y<\/mi><mo>=<\/mo><mn>10<\/mn><mo>+<\/mo><mn>12<\/mn><mo>\u21d2<\/mo><mn>11<\/mn><mi>x<\/mi><mo>=<\/mo><mn>22<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3x + 4y + 8x &#8211; 4y = 10 + 12 \\Rightarrow 11x = 22 \\Rightarrow x = 2<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord\">11<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">22<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/li>\n<li>Substitute <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 2<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span> in <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x &#8211; y = 3<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>4<\/mn><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>3<\/mn><mo>\u21d2<\/mo><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4 &#8211; y = 3 \\Rightarrow y = 1<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/li>\n<li>Solution: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 2, y = 1<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + 3y = 6<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span> <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x &#8211; 3y = 12<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><\/span><\/span><\/span><\/p>\n<ul>\n<li>Add both equations: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>6<\/mn><mo>+<\/mo><mn>12<\/mn><mo>\u21d2<\/mo><mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>18<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + 3y + 2x &#8211; 3y = 6 + 12 \\Rightarrow 3x = 18 \\Rightarrow x = 6<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">12<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">18<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span><\/li>\n<li>Substitute <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 6<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span> in <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + 3y = 6<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>6<\/mn><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>6<\/mn><mo>\u21d2<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><mo>\u21d2<\/mo><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">6 + 3y = 6 \\Rightarrow 3y = 0 \\Rightarrow y = 0<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">6<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/li>\n<li>Solution: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>6<\/mn><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 6, y = 0<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<h3><span class=\"ez-toc-section\" id=\"Worksheet_4_Word_Problems\"><\/span>Worksheet 4: Word Problems<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Instructions:<\/strong> Solve the following word problems using the concept of linear equations in two variables.<\/p>\n<ol>\n<li>The sum of two numbers is 14, and their difference is 2. Find the numbers.<\/li>\n<li>The cost of 5 pens and 3 pencils is \u20b923. The cost of 2 pens and 2 pencils is \u20b910. Find the cost of each pen and pencil.<\/li>\n<\/ol>\n<p><strong>Solutions:<\/strong><\/p>\n<ol>\n<li>Let the numbers be <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span>.<\/p>\n<ul>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + y = 14<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">14<\/span><\/span><\/span><\/span><\/li>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x &#8211; y = 2<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/li>\n<li>Add both equations: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mn>16<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x = 16 \\Rightarrow x = 8<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">16<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><\/span><\/span><\/span><\/li>\n<li>Substitute <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>8<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 8<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><\/span><\/span><\/span> in <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + y = 14<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">14<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>8<\/mn><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>14<\/mn><mo>\u21d2<\/mo><mi>y<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">8 + y = 14 \\Rightarrow y = 6<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">8<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">14<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">6<\/span><\/span><\/span><\/span><\/li>\n<li>Solution: The numbers are 8 and 6.<\/li>\n<\/ul>\n<\/li>\n<li>Let the cost of one pen be <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span> and one pencil be <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span>.<\/p>\n<ul>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>23<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5x + 3y = 23<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">23<\/span><\/span><\/span><\/span><\/li>\n<li><span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>y<\/mi><mo>=<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + 2y = 10<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">10<\/span><\/span><\/span><\/span><\/li>\n<li>Simplify second equation: <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + y = 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span><\/li>\n<li>Solve <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + y = 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span> for <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y = 5 &#8211; x<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/li>\n<li>Substitute <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y = 5 &#8211; x<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span> in <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mn>23<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5x + 3y = 23<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">23<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>23<\/mn><mo>\u21d2<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/mn><mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>23<\/mn><mo>\u21d2<\/mo><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mn>8<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5x + 3(5 &#8211; x) = 23 \\Rightarrow 5x + 15 &#8211; 3x = 23 \\Rightarrow 2x = 8 \\Rightarrow x = 4<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">5<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">23<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">15<\/span><span class=\"mbin\">\u2212<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">23<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">8<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span><\/li>\n<li>Substitute <span class=\"katex\"><span class=\"katex-mathml\"><br \/>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 4<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span> in <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + y = 5<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><\/span><\/span><\/span>: <span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>4<\/mn><mo>+<\/mo><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><mo>\u21d2<\/mo><mi>y<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4 + y = 5 \\Rightarrow y = 1<\/annotation><\/semantics><\/math>\n<p><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/li>\n<li>Solution: The cost of each pen is \u20b94 and each pencil is \u20b91.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Class 10 Maths Worksheets for Chapter 3 (Pair of Linear Equations in Two Variables) are available here online with answers. [&hellip;]<\/p>\n","protected":false},"author":56,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Class 10 Math","_yoast_wpseo_title":"Worksheet for Class 10 Math Pair of Linear Equations in Two Variables","_yoast_wpseo_metadesc":"MCQs for Class 10 Math Pair of Linear Equations in Two Variables are available at IL with answers and detailed explanations for each question","custom_permalink":"worksheets-class-10-maths\/pair-of-linear-equations-in-two-variables\/"},"categories":[11033],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Worksheet for Class 10 Math Pair of Linear Equations in Two Variables<\/title>\n<meta name=\"description\" content=\"MCQs for Class 10 Math Pair of Linear Equations in Two Variables are available at IL with answers and detailed explanations 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