{"id":9102,"date":"2026-06-01T21:33:48","date_gmt":"2026-06-01T21:33:48","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9102"},"modified":"2026-06-01T21:33:48","modified_gmt":"2026-06-01T21:33:48","slug":"introduction-to-linear-equation","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/introduction-to-linear-equation\/","title":{"rendered":"Introduction To Linear Equation"},"content":{"rendered":"<p><strong>Unit: <\/strong><strong>Algebra &#8211; 1<\/strong><\/p>\n<p><strong>Chapter: <\/strong><strong>Introduction to Two variables<\/strong><\/p>\n<p><em>Reference: &#8211; Introduction to Linear Equations in Two Variables, General Form, Solutions of an Equation, Graph of a Linear Equation, Infinite Solutions, Intercepts, Standard Form, Slope-Intercept Form, Applications, Solved Examples, Odd-One-Out Problems, Common Mistakes<\/em><\/p>\n<p><strong>After studying this chapter, you should be able to understand:<\/strong><\/p>\n<ul>\n<li><em>Introduction to Linear Equation in Two Variable<\/em><\/li>\n<li><em>General Form &amp; Solution Concepts<\/em><\/li>\n<li><em>Infinite Solutions Property<\/em><\/li>\n<li><em>Graphical Representation<\/em><\/li>\n<\/ul>\n<p><strong>Introduction to Linear Equations in Two Variables<\/strong><\/p>\n<p><strong><u>Definition<\/u><\/strong><\/p>\n<p>A linear equation in two variables is an equation that can be written in the form&nbsp;ax + by + c = 0&nbsp;(or&nbsp;ax + by = c), where a, b, and c are real numbers, and&nbsp;a and b are not both zero. The variables (usually x and y) have the highest power of 1.<\/p>\n<p>When we study linear equations in two variables, we essentially ask: &quot;What are all the pairs of (x, y) that satisfy this equation?&quot;<\/p>\n<p>Unlike equations in one variable (which have a single solution), equations in two variables have&nbsp;infinitely many solutions.<\/p>\n<p><strong><u>Importance of Linear Equations in Two Variables<\/u><\/strong><\/p>\n<ul>\n<li>Models&rsquo; real-world relationships between two quantities (cost and quantity, distance and time, etc.)<\/li>\n<li>Foundation for graphing lines on a coordinate plane<\/li>\n<li>Essential for solving systems of equations<\/li>\n<li>Used extensively in economics, physics, and engineering<\/li>\n<\/ul>\n<p><strong>Example<\/strong><\/p>\n<p><strong>Equation:<\/strong>&nbsp;x + y = 5<br \/>\n<strong>Some solutions:<\/strong>&nbsp;(1,4), (2,3), (3,2), (4,1), (0,5), (5,0), (2.5, 2.5)<br \/>\n<strong>Common Property:<\/strong>&nbsp;In every solution, the sum of x and y is 5. So, if we are given (6, -1), it also satisfies because 6 + (-1) = 5.<\/p>\n<p><strong><u>Subtopics<\/u><\/strong><\/p>\n<p><strong>1. General Form<\/strong><\/p>\n<p>The general form of a linear equation in two variables is&nbsp;ax + by + c = 0, where a, b, c are real numbers, and at least one of a or b is non-zero.<\/p>\n<p><strong>Examples of General Form:<\/strong><\/p>\n<ul>\n<li>2x + 3y &#8211; 6 = 0 (here a=2, b=3, c=-6)<\/li>\n<li>x &#8211; y = 4 can be written as x &#8211; y &#8211; 4 = 0 (a=1, b=-1, c=-4)<\/li>\n<li>y = 2x + 1 can be written as -2x + y &#8211; 1 = 0 (a=-2, b=1, c=-1)<\/li>\n<li>x = 5 can be written as x &#8211; 5 = 0 (a=1, b=0, c=-5) &rarr; vertical line<\/li>\n<li>y = -3 can be written as y + 3 = 0 (a=0, b=1, c=3) &rarr; horizontal line<\/li>\n<\/ul>\n<p><strong>Special Cases:<\/strong><\/p>\n<ul>\n<li>If b = 0, the equation becomes ax + c = 0 &rarr; x = constant (vertical line)<\/li>\n<li>If a = 0, the equation becomes by + c = 0 &rarr; y = constant (horizontal line)<\/li>\n<\/ul>\n<p><strong>2. Solution of a Linear Equation in Two Variables<\/strong><\/p>\n<p>A&nbsp;solution&nbsp;is an ordered pair (x, y) that satisfies the equation (makes LHS equal to RHS).<\/p>\n<p><strong>Key Property:<\/strong>&nbsp;A linear equation in two variables has&nbsp;infinitely many solutions.<\/p>\n<p><strong>Example of Finding Solutions:<\/strong>&nbsp;For the equation 2x + y = 7<\/p>\n<p>If we choose x = 0, then y = 7 &rarr; solution (0, 7)<br \/>\nIf we choose x = 1, then y = 5 &rarr; solution (1, 5)<br \/>\nIf we choose x = 2, then y = 3 &rarr; solution (2, 3)<br \/>\nIf we choose x = 3, then y = 1 &rarr; solution (3, 1)<br \/>\nIf we choose x = 4, then y = -1 &rarr; solution (4, -1)<br \/>\nIf we choose x = -1, then y = 9 &rarr; solution (-1, 9)<br \/>\nIf we choose x = 2.5, then y = 2 &rarr; solution (2.5, 2)<\/p>\n<p><strong>Note:<\/strong>&nbsp;You can choose&nbsp;any&nbsp;value for one variable and calculate the corresponding value of the other variable.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>3. Graphical Representation<\/strong><\/p>\n<p>When we plot all solutions of a linear equation in two variables on a coordinate plane, they form a&nbsp;straight line. This is why it&#39;s called a &quot;linear&quot; equation.<\/p>\n<p><strong>Key Points about the Graph:<\/strong><\/p>\n<ul>\n<li>Every point on the line is a solution.<\/li>\n<li>Any point not on the line is not a solution.<\/li>\n<li>A line is determined by&nbsp;just two points.<\/li>\n<\/ul>\n<p><strong>Steps to Graph a Linear Equation:<\/strong><\/p>\n<ul>\n<li>Step 1: Find two solutions (ordered pairs)<\/li>\n<li>Step 2: Plot them on the coordinate plane<\/li>\n<li>Step 3: Draw a straight line through them<\/li>\n<\/ul>\n<p><strong>Example &ndash; Graphing x &#8211; y = 2:<\/strong><\/p>\n<p>One solution: if x = 0, then y = -2 &rarr; point (0, -2)<br \/>\nAnother solution: if x = 2, then y = 0 &rarr; point (2, 0)<br \/>\nPlot (0,-2) and (2,0) and draw the line through them.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4. Standard Forms of Linear Equations<\/strong><\/p>\n<p>Linear equations in two variables can be written in different forms, each useful for different purposes.<\/p>\n<p><strong>Standard Form:<\/strong>&nbsp;ax + by = c<br \/>\nExample: 3x + 4y = 12<\/p>\n<p><strong>Slope-Intercept Form:<\/strong>&nbsp;y = mx + b<br \/>\nHere, m = slope (steepness of the line) and b = y-intercept (where the line crosses the y-axis)<br \/>\nExample: y = 2x + 3 &rarr; slope = 2, y-intercept = 3<\/p>\n<p><strong>Intercept Form:<\/strong>&nbsp;x\/a + y\/b = 1<br \/>\nHere, a = x-intercept and b = y-intercept<br \/>\nExample: x\/4 + y\/3 = 1 &rarr; x-intercept = 4, y-intercept = 3<\/p>\n<p>&nbsp;<\/p>\n<p><strong>5. Intercepts<\/strong><\/p>\n<p>Intercepts are the points where the line crosses the axes.<\/p>\n<p><strong>x-intercept:<\/strong>&nbsp;Set y = 0, then solve for x. The coordinates are (a, 0).<\/p>\n<p><strong>y-intercept:<\/strong>&nbsp;Set x = 0, then solve for y. The coordinates are (0, b).<\/p>\n<p>Example &ndash; Finding Intercepts for 2x + 3y = 12:<\/p>\n<p>To find x-intercept: set y = 0 &rarr; 2x = 12 &rarr; x = 6 &rarr; point (6, 0)<\/p>\n<p>To find y-intercept: set x = 0 &rarr; 3y = 12 &rarr; y = 4 &rarr; point (0, 4)<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Solved Examples<\/strong><\/p>\n<p><strong>Example 1:<\/strong>&nbsp;Check if (2, 3) is a solution of 2x &#8211; y = 1.<\/p>\n<p><strong>Solution:<\/strong>&nbsp;LHS = 2(2) &#8211; 3 = 4 &#8211; 3 = 1 = RHS. Yes, it is a solution.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 2:<\/strong>&nbsp;Find four solutions of the equation x + 2y = 8.<\/p>\n<p><strong>Solution:<\/strong>&nbsp;Choose any four values for x and find y.<\/p>\n<p>If x = 0, then y = (8 &#8211; 0)\/2 = 4 &rarr; solution (0, 4)<br \/>\nIf x = 2, then y = (8 &#8211; 2)\/2 = 3 &rarr; solution (2, 3)<br \/>\nIf x = 4, then y = (8 &#8211; 4)\/2 = 2 &rarr; solution (4, 2)<br \/>\nIf x = 6, then y = (8 &#8211; 6)\/2 = 1 &rarr; solution (6, 1)<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 3:<\/strong>&nbsp;Write the equation 3x + 2y = 6 in slope-intercept form.<\/p>\n<p><strong>Solution:<\/strong>&nbsp;Solve for y.<br \/>\n3x + 2y = 6<br \/>\n2y = -3x + 6<br \/>\ny = (-3\/2)x + 3<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 4:<\/strong>&nbsp;Find the x-intercept and y-intercept of 5x &#8211; 3y = 15.<\/p>\n<p><strong>Solution:<\/strong><br \/>\nx-intercept: set y = 0 &rarr; 5x = 15 &rarr; x = 3 &rarr; point (3, 0)<br \/>\ny-intercept: set x = 0 &rarr; -3y = 15 &rarr; y = -5 &rarr; point (0, -5)<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 5 &ndash; Odd One Out (Ordered Pairs):<\/strong><\/p>\n<p><strong>Examine the five ordered pairs below. Exactly one is NOT a solution of the equation 2x + y = 10. Identify it.<\/strong><br \/>\nPairs: (2, 6), (3, 4), (4, 2), (5, 0), (1, 7)<\/p>\n<p><strong>Solution:<\/strong>&nbsp;Check each pair in the equation 2x + y = 10.<\/p>\n<p>For (2, 6): 2(2) + 6 = 4 + 6 = 10 \u2713<br \/>\nFor (3, 4): 2(3) + 4 = 6 + 4 = 10 \u2713<br \/>\nFor (4, 2): 2(4) + 2 = 8 + 2 = 10 \u2713<br \/>\nFor (5, 0): 2(5) + 0 = 10 + 0 = 10 \u2713<br \/>\nFor (1, 7): 2(1) + 7 = 2 + 7 = 9 &ne; 10 \u2717<\/p>\n<p><strong>Three reasons why (1, 7) is the odd one out:<\/strong><\/p>\n<p><strong>(A)<\/strong>&nbsp;It does not satisfy the equation 2x + y = 10 (gives 9 instead of 10).<br \/>\n<strong>(B)<\/strong>&nbsp;In all other pairs, the sum 2x + y equals 10; in (1,7), it equals 9.<br \/>\n<strong>(C)<\/strong>&nbsp;On a graph, (1,7) lies on a different line (2x + y = 9), not on the line 2x + y = 10.<\/p>\n<p><strong>Conclusion:<\/strong>&nbsp;(1, 7) is the odd one out.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Example 6 &ndash; Odd One Out (Equations):<\/strong><\/p>\n<p><strong>Examine the five equations below. Exactly one is NOT a linear equation in two variables. Identify it.<\/strong><br \/>\nEquations: 2x + 3y = 12, y = 4x &#8211; 5, x&sup2; + y = 9, 3x &#8211; y = 7, x = 2y + 1<\/p>\n<p><strong>Solution:<\/strong><br \/>\n2x + 3y = 12 &rarr; linear (powers of x and y are 1)<br \/>\ny = 4x &#8211; 5 &rarr; linear (powers of x and y are 1)<br \/>\nx&sup2; + y = 9 &rarr; NOT linear (x has power 2, which is quadratic)<br \/>\n3x &#8211; y = 7 &rarr; linear (powers of x and y are 1)<br \/>\nx = 2y + 1 &rarr; linear (powers of x and y are 1)<\/p>\n<p><strong>Three reasons why x&sup2; + y = 9 is the odd one out:<\/strong><\/p>\n<p><strong>(A)<\/strong>&nbsp;Its degree is 2 (because of x&sup2;), while all others have degree 1.<br \/>\n<strong>(B)<\/strong>&nbsp;Its graph is a parabola (curved), while all others graph as straight lines.<br \/>\n<strong>(C)<\/strong>&nbsp;The variable x has exponent 2, violating the definition of a linear equation.<\/p>\n<p><strong>Conclusion:<\/strong>&nbsp;x&sup2; + y = 9 is the odd one out.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Common Mistakes to Avoid<\/strong><\/p>\n<p><strong>Mistake 1 &ndash; Thinking only one solution exists<\/strong><br \/>\nWhy it&#39;s wrong: A linear equation in two variables has infinitely many solutions. For any value of x, we can find a corresponding y.<br \/>\nCorrect understanding: Solutions are infinite; we can only list a few.<\/p>\n<p><strong>Mistake 2 &ndash; Confusing (x,y) with (y,x)<\/strong><br \/>\nWhy it&#39;s wrong: The order matters. (2,3) and (3,2) are different points unless x equals y.<br \/>\nCorrect understanding: In an ordered pair, the first number is always x and the second is always y.<\/p>\n<p><strong>Mistake 3 &ndash; Forgetting negative signs when finding intercepts<\/strong><br \/>\nWhy it&#39;s wrong: Setting x=0 gives y-intercept, but solving incorrectly may give the wrong sign.<br \/>\nCorrect understanding: Carefully solve the equation after substituting zero.<\/p>\n<p><strong>Mistake 4 &ndash; Assuming line is vertical or horizontal without checking<\/strong><br \/>\nWhy it&#39;s wrong: The equation ax + by = c is vertical only if b = 0, and horizontal only if a = 0.<br \/>\nCorrect understanding: Check the coefficients before deciding.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Quick Reference Summary<\/strong><\/p>\n<p><strong>General Form:<\/strong>&nbsp;ax + by + c = 0 (a and b not both zero)<\/p>\n<p><strong>Solutions:<\/strong>&nbsp;Infinitely many ordered pairs (x, y) that satisfy the equation<\/p>\n<p><strong>Graph:<\/strong>&nbsp;Always a straight line<\/p>\n<p><strong>x-intercept:<\/strong>&nbsp;Set y = 0, solve for x &rarr; point (a, 0)<\/p>\n<p><strong>y-intercept:<\/strong>&nbsp;Set x = 0, solve for y &rarr; point (0, b)<\/p>\n<p><strong>Slope-Intercept Form:<\/strong>&nbsp;y = mx + b (m = slope, b = y-intercept)<\/p>\n<p><strong>Standard Form:<\/strong>&nbsp;ax + by = c<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Unit: Algebra &#8211; 1 Chapter: Introduction to Two variables Reference: &#8211; Introduction to Linear Equations in Two Variables, General Form, Solutions of an Equation, Graph of a Linear Equation, Infinite Solutions, Intercepts, Standard Form, Slope-Intercept Form, Applications, Solved Examples, Odd-One-Out Problems, Common Mistakes After studying this chapter, you should be able to understand: Introduction to [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[593],"tags":[],"class_list":["post-9102","post","type-post","status-publish","format-standard","hentry","category-grade-8"],"_links":{"self":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9102","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/comments?post=9102"}],"version-history":[{"count":0,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/posts\/9102\/revisions"}],"wp:attachment":[{"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/media?parent=9102"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/categories?post=9102"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kapdec.com\/help\/wp-json\/wp\/v2\/tags?post=9102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}