{"id":2846,"date":"2024-02-09T19:35:08","date_gmt":"2024-02-09T19:35:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=2846"},"modified":"2026-03-09T16:37:00","modified_gmt":"2026-03-09T16:37:00","slug":"6-1-1-solving-linear-equations-in-one-variable","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/6-1-1-solving-linear-equations-in-one-variable\/","title":{"raw":"6.1.1 Solving Linear Equations in One Variable","rendered":"6.1.1 Solving Linear Equations in One Variable"},"content":{"raw":"<nav style=\"background-color: #007fab;\" role=\"navigation\">\r\n<div class=\"book-title-wrapper\">\r\n<div class=\"bombadil-logo\"><a href=\"https:\/\/courses.lumenlearning.com\/\"><img src=\"https:\/\/courses.lumenlearning.com\/boundless-algebra\/wp-content\/themes\/bombadil\/assets\/images\/LumenOnDark-150x69.png\" alt=\"Lumen\" \/><\/a><\/div>\r\n<\/div>\r\n<\/nav>\r\n<div id=\"content\" role=\"main\">\r\n<h1 class=\"entry-title\">Solving Linear Equations<\/h1>\r\n<div id=\"post-902\" class=\"standard post-902 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div id=\"wpipa-1404-container\" class=\"wpipa-container wpipa-align-center\" data-id=\"1404\" data-variation=\"none\">\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Determine whether or not an equation in one variable is defined as a linear equation<\/li>\r\n \t<li>Use the addition and subtraction property to solve a linear equation<\/li>\r\n \t<li>Use the multiplication and division property to solve a linear equation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h1>KEY WORDS<\/h1>\r\n<ul>\r\n \t<li><strong>Equality<\/strong>: The state of two or more entities having the same value.<\/li>\r\n \t<li><strong>Equation<\/strong>: a statement that two expressions are equal.<\/li>\r\n \t<li><strong>Solution<\/strong>: a number or numbers that make an equation true.<\/li>\r\n \t<li><strong>Equivalent equations:<\/strong> two or more equations that have identical solutions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h2>Linear Equations<\/h2>\r\nAn equation with just <strong>one variable<\/strong> is said to be\u00a0<em><strong>linear<\/strong><\/em> when the highest power on the variable is [latex]1[\/latex]. Remember that [latex]{x}^{1}[\/latex] is equivalent to [latex]x[\/latex], so any equation that can be simplified to [latex]ax+b=c[\/latex] (where [latex]a,b,c[\/latex] are real numbers) is a <strong>linear equation in one variable<\/strong>. For example, the equation [latex]2x+3=7, \\;5x=7,\\; -\\frac{3}{4}x+\\frac{7}{3}=\\frac{6}{5}[\/latex] are all linear equations because the highest power of [latex]x[\/latex] is [latex]1[\/latex]. On the other hand, the equation [latex]{x}^3+2{x}^2=4x-2[\/latex] is NOT linear because the highest power of\u00a0[latex]x[\/latex] is\u00a0[latex]3[\/latex], not\u00a0[latex]1[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nDetermine whether the equation is linear:\r\n<ul>\r\n \t<li>a)\u00a0[latex]3x-7=6x+2[\/latex]<\/li>\r\n \t<li>b)\u00a0[latex]{x}^{3}-5{x}^{2}=5x+6[\/latex]<\/li>\r\n \t<li>c)\u00a0[latex]x(x+6)=8[\/latex]<\/li>\r\n \t<li>d)\u00a0[latex]x-\\frac{3}{8} =\\frac{1}{2}[\/latex]<\/li>\r\n<\/ul>\r\n<h4><strong>Solution:<\/strong><\/h4>\r\n<table id=\"eip-id1168466426761\" class=\"unnumbered unstyled\" summary=\"The top line says 4y plus 3 equals 8y. Beside this is \">\r\n<tbody>\r\n<tr>\r\n<th>Equation<\/th>\r\n<th>Explanation<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>a) [latex]3x-7=6x+2[\/latex]<\/td>\r\n<td>Yes, it is linear since the highest power of [latex]x[\/latex] is\u00a0[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>b) [latex]{x}^{3}-5{x}^{2}=5x+6[\/latex]<\/td>\r\n<td>No, it is NOT linear since the highest power of\u00a0[latex]x[\/latex] is\u00a0[latex]3[\/latex], not\u00a0[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>c) [latex]x(x+6)=8[\/latex]<\/td>\r\n<td>Use the distributive property to simplify the left side of the equation:\u00a0[latex]{x}^{2}+6x=8[\/latex]. Now we can see that the equation is NOT linear\u00a0since the highest power of\u00a0[latex]x[\/latex] is\u00a0[latex]2[\/latex], not\u00a0[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>d) [latex]x-\\frac{3}{8} =\\frac{1}{2}[\/latex]<\/td>\r\n<td>Yes, it is linear since the highest power of [latex]x[\/latex] is\u00a0[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\nDetermine whether the equation [latex]6x-7=8{x}^{2}+5[\/latex] is linear.\r\n\r\n[reveal-answer q=\"H411001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"H411001\"]\r\n\r\nSolution:\r\n\r\n[latex]6x-7=8{x}^{2}+5[\/latex] is NOT linear since the highest power of [latex]x[\/latex] is [latex]2[\/latex], not [latex]1[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\nDetermine whether the equation [latex]\\frac{4}{5}x+6=\\frac{-7}{3}x[\/latex] is linear.\r\n\r\n[reveal-answer q=\"H411002\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"H411002\"]\r\n\r\nSolution:\r\n\r\n[latex]\\frac{4}{5}x+6=\\frac{-7}{3}x[\/latex] is linear since the highest power of [latex]x[\/latex] is [latex]1[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>The Addition and Subtraction Property of Equality<\/h2>\r\nWhen an equation contains a variable such as [latex]x[\/latex], this variable is considered an <strong><em>unknown<\/em><\/strong> value. In many cases, we can find the possible values for [latex]x[\/latex] that would make the equation true. We can\u00a0<em><strong>solve\u00a0<\/strong><\/em>the equation for [latex]x[\/latex].\r\n\r\nFor some equations like [latex]x + 3 =5[\/latex] it is easy to guess the solution: the only possible value of [latex]x[\/latex] is [latex]2[\/latex], because [latex]2 + 3 = 5[\/latex].\u00a0However, it becomes useful to have a process for finding solutions for unknowns as problems become more complex.\r\n\r\nIn order to solve an equation, we need to\u00a0<em><b style=\"color: #000000; font-size: 1rem; orphans: 1; text-align: initial;\">isolate the variable<\/b><\/em><span style=\"color: #000000; font-size: 1rem; font-weight: normal; orphans: 1; text-align: initial;\">. Isolating the variable means rewriting an <em><strong>equivalent equation<\/strong><\/em> in which the variable is on one side of the equation and everything else is on the other side of the equation. The variable is an <em><strong>unknown<\/strong><\/em> quantity whose value we are trying to find.\u00a0We have a solution when we reorganize the equation into the form [latex]x[\/latex] = a constant.<\/span>\r\n\r\n<\/div>\r\n<div class=\"entry-content\"><span style=\"color: #000000; font-size: 1rem; orphans: 1; text-align: initial;\">An equation is two expressions that are set equal to each other. If we perform an operation (+, \u2013, \u00b7, \u00f7, etc) to one side of the equation, it will upset the balance of the equation. Our goal is to keep the balance of the equation intact. This can be likened to a balance scale where the equation [latex]x+3=5[\/latex] is illustrated:<\/span><\/div>\r\n<div><\/div>\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-1853 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/09\/26203113\/Scale-equation-300x282.png\" alt=\"Scale\" width=\"300\" height=\"282\" \/><\/p>\r\nThe second we add or subtract a \"weight\" to one side of the scale, it will become unbalanced. In order to right that balance, we must add or subtract the same \"weight\" to the other side of the sale.\r\n\r\nTo solve this equation for [latex]x[\/latex], we need to isolate [latex]x[\/latex] by removing the [latex]+3[\/latex] from the left side of the equation. We do this this by \"undoing\" the addition of\u00a0[latex]+3[\/latex] by subtracting\u00a0[latex]3[\/latex].\u00a0 In other words we use the inverse operation of addition, which is subtraction. But if we subtract\u00a0[latex]3[\/latex] from the left side of the equation, we must also subtract\u00a0[latex]3[\/latex] from the right side to keep the equation balanced.\r\n<p style=\"text-align: center;\">[latex]x+3\\;\\;\\;\\;\\;\\;=5 \\\\ x+3\\color{blue}{-3}=5\\color{blue}{-3} \\\\ x\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\,=2[\/latex]<\/p>\r\nWe can verify that [latex]x=2[\/latex] is indeed the solution by substituting\u00a0[latex]2[\/latex] in for\u00a0[latex]x[\/latex] in the original equation: [latex]x+3=5 \\\\ 2+3=5 \\\\ 5=5[\/latex] Since this is true,\u00a0[latex]x=2[\/latex] is the solution of\u00a0[latex]x+3=5[\/latex].\r\n\r\n<span style=\"color: #000000;\">If the equation is [latex]x-5=9[\/latex], we isolate the variable by adding [latex]5[\/latex] to both sides of the equation. This is because, to isolate [latex]x[\/latex] we must \"undo\" subtract [latex]5[\/latex] by adding [latex]5[\/latex]:\u00a0<\/span>\r\n\r\n[latex]x-5=9 \\\\ x-5\\color{blue}{+5}=9\\color{blue}{+5} \\\\ x\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\,=14[\/latex]\r\n\r\nAgain, we can verify the solution:\u00a0[latex]x-5\\;\\;\\;\\;\\;\\;=9 \\\\ 14 - 5 = 9 \\\\ 9=9[\/latex] True.\r\n\r\nWhen we add or subtract the same term to both sides of an equation, we get an <em><strong>equivalent equation<\/strong><\/em>. <strong>Equivalent equations<\/strong> are two or more equations that have identical solutions. This leads us to the the <em><strong>Addition and Subtraction Property of Equality<\/strong><\/em>.\r\n<div class=\"textbox shaded\">\r\n<h3 id=\"fs-id1166490863495\"><strong>Addition &amp; subtraction Property of Equality<\/strong><\/h3>\r\n<span style=\"color: #000000;\">For all real numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex] and [latex]a-c=b-c[\/latex].<\/span>\r\n\r\n<span style=\"color: #000000;\">Adding or subtracting the same term to both sides of an equation will result in an <strong><em>equivalent equation<\/em><\/strong>.<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\n<span style=\"color: #000000;\">Solve: [latex]x+11=-3[\/latex]<\/span>\r\n<h4><span style=\"color: #000000;\">Solution:<\/span><\/h4>\r\n<span style=\"color: #000000;\">To isolate [latex]x[\/latex], we undo the addition of [latex]11[\/latex] by using the Subtraction Property of Equality.<\/span>\r\n<table id=\"eip-id1168468291100\" class=\"unnumbered unstyled\" style=\"height: 90px;\" summary=\"The top line says x plus 11 equals negative 3. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 349.656px;\" colspan=\"3\">Solving [latex]x+11=-3[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"height: 11px;\" colspan=\"2\"><\/td>\r\n<td style=\"height: 11px; width: 206.656px;\">[latex]x+11=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 23px;\">\r\n<td style=\"height: 23px; width: 349.656px;\" colspan=\"2\">Subtract 11 from each side to \"undo\" the addition.<\/td>\r\n<td style=\"height: 23px; width: 206.656px;\">[latex]x+11\\color{red}{-11}=-3\\color{red}{-11}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"height: 11px; width: 349.656px;\" colspan=\"2\">Simplify.<\/td>\r\n<td style=\"height: 11px; width: 206.656px;\">[latex]x=-14[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"height: 11px; width: 121.656px;\">Check:<\/td>\r\n<td style=\"height: 11px; width: 216.656px;\">[latex]x+11=-3[\/latex]<\/td>\r\n<td style=\"height: 11px; width: 206.656px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 23px;\">\r\n<td style=\"height: 23px; width: 121.656px;\">Substitute [latex]x=-14[\/latex] .<\/td>\r\n<td style=\"height: 23px; width: 216.656px;\">[latex]\\color{red}{-14}+11\\stackrel{\\text{?}}{=}-3[\/latex]<\/td>\r\n<td style=\"height: 23px; width: 206.656px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px;\">\r\n<td style=\"height: 11px; width: 121.656px;\"><\/td>\r\n<td style=\"height: 11px; width: 216.656px;\">[latex]-3=-3\\quad\\checkmark[\/latex]<\/td>\r\n<td style=\"height: 11px; width: 206.656px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span style=\"color: #000000;\">Since [latex]x=-14[\/latex] makes [latex]x+11=-3[\/latex] a true statement, we know that it is a solution to the equation.<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141721&amp;theme=oea&amp;iframe_resize_id=mom200[\/embed]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\n<span style=\"color: #000000;\">Solve: [latex]m - 4=-5[\/latex]<\/span>\r\n<h4><span style=\"color: #000000;\">Solution:<\/span><\/h4>\r\n<table id=\"eip-id1168467157298\" class=\"unnumbered unstyled\" summary=\"The first line says m minus 4 equals negative 5. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<th colspan=\"3\">Solving [latex]m-4=-5[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>[latex]m-4=-5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Add 4 to each side to \"undo\" the subtraction.<\/td>\r\n<td>[latex]m-4\\color{red}{+4}=-5\\color{red}{+4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]m=-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]m-4=-5[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]m=-1[\/latex] .<\/td>\r\n<td>[latex]\\color{red}{-1}+4\\stackrel{\\text{?}}{=}-5[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-5=-5\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>The solution to [latex]m - 4=-5[\/latex] is [latex]m=-1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141723&amp;theme=oea&amp;iframe_resize_id=mom27[\/embed]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/yqdlj0lv7Cc\r\n<div id=\"post-902\" class=\"standard post-902 chapter type-chapter status-publish hentry\"><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-1.docx\">Transcript-6.1.1-1<\/a><\/div>\r\n<h2>Linear Equations Containing Fractions<\/h2>\r\n<span style=\"color: #000000;\">It is not uncommon to encounter equations that contain fractions; therefore, in the following examples, we will demonstrate how to use the addition property of equality to solve an equation with fractions.<\/span>\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\n<span style=\"color: #000000;\">Solve: [latex]n-\\frac{3}{8} =\\frac{1}{2}[\/latex]<\/span>\r\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\r\n<table id=\"eip-id1168468255179\" class=\"unnumbered unstyled\" style=\"width: 602px;\" summary=\"The first line says n minus 3 eighths equals one half. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 313.391px;\" colspan=\"3\">Solving [latex]n-\\frac{3}{8}=\\frac{1}{2}[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td style=\"width: 255.609px;\">[latex]n-\\frac{3}{8} =\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 313.391px;\" colspan=\"2\">Use the Addition Property of Equality.<\/td>\r\n<td style=\"width: 255.609px;\">[latex]n-\\frac{3}{8}\\color{red}{+\\frac{3}{8}} =\\frac{1}{2}\\color{red}{+\\frac{3}{8}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 313.391px;\" colspan=\"2\">Find the LCD to add the fractions on the right.<\/td>\r\n<td style=\"width: 255.609px;\">[latex]n-\\frac{3}{8} +\\frac{3}{8} =\\frac{4}{8} +\\frac{3}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 313.391px;\" colspan=\"2\">Simplify.<\/td>\r\n<td style=\"width: 255.609px;\">[latex]n=\\frac{7}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 142px;\">Check:<\/td>\r\n<td style=\"width: 171.391px;\">[latex]n-\\frac{3}{8} =\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 255.609px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 142px;\">Substitute [latex]n=\\color{red}{\\frac{7}{8}}[\/latex]<\/td>\r\n<td style=\"width: 171.391px;\">[latex]\\color{red}{\\frac{7}{8}} -\\frac{3}{8}\\stackrel{\\text{?}}{=}\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 255.609px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 142px;\">Subtract.<\/td>\r\n<td style=\"width: 171.391px;\">[latex]\\frac{4}{8}\\stackrel{\\text{?}}{=}\\frac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 255.609px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 142px;\">Simplify.<\/td>\r\n<td style=\"width: 171.391px;\">[latex]\\frac{1}{2}\\normalsize =\\frac{1}{2}\\normalsize\\quad\\checkmark[\/latex]<\/td>\r\n<td style=\"width: 255.609px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 142px;\">The solution checks.<\/td>\r\n<td style=\"width: 171.391px;\"><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3><span style=\"color: #000000;\">TRY<\/span>\u00a0IT<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141732&amp;theme=oea&amp;iframe_resize_id=mom25[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<span style=\"color: #000000;\">Watch this video for more examples of solving equations that include fractions and require addition or subtraction.<\/span>\r\n\r\nhttps:\/\/youtu.be\/KmOvCakGEgM\r\n<h2><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-2-1.docx\">Transcript-6.1.1-2<\/a><\/h2>\r\n<h2>Linear Equations Containing Decimals<\/h2>\r\nDecimals will be encountered any time money or metric measurements are used.\r\n<div class=\"textbox exercises\">\r\n<h3>eXAMPLE<\/h3>\r\n<span style=\"color: #000000;\">Solve [latex]a - 3.7=4.3[\/latex]<\/span>\r\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\r\n<table id=\"eip-id1168468751492\" class=\"unnumbered unstyled\" style=\"width: 730px;\" summary=\"The top line says a minus 3.7 equals 4.3. The next line says \">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 516.391px;\" colspan=\"3\">Solving [latex]a-3.7=4.3[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td style=\"width: 180.609px;\">[latex]a-3.7=4.3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 516.391px;\" colspan=\"2\">Use the Addition Property of Equality.<\/td>\r\n<td style=\"width: 180.609px;\">[latex]a-3.7\\color{red}{+3.7}=4.3\\color{red}{+3.7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 516.391px;\" colspan=\"2\">Add.<\/td>\r\n<td style=\"width: 180.609px;\">[latex]a=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 176px;\">Check:<\/td>\r\n<td style=\"width: 340.391px;\">[latex]a-3.7=4.3[\/latex]<\/td>\r\n<td style=\"width: 180.609px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 176px;\">Substitute [latex]a=8[\/latex] .<\/td>\r\n<td style=\"width: 340.391px;\">[latex]\\color{red}{8}-3.7\\stackrel{\\text{?}}{=}4.3[\/latex]<\/td>\r\n<td style=\"width: 180.609px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 176px;\">Simplify.<\/td>\r\n<td style=\"width: 340.391px;\">[latex]4.3=4.3\\quad\\checkmark[\/latex]<\/td>\r\n<td style=\"width: 180.609px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 176px;\">The solution checks.<\/td>\r\n<td style=\"width: 340.391px;\"><\/td>\r\n<td style=\"width: 180.609px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141736&amp;theme=oea&amp;iframe_resize_id=mom29[\/embed]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/a6YYJN_bHKs\r\n<h2><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-3.docx\">Transcript-6.1.1-3<\/a><\/h2>\r\n<h2 id=\"title2\">The Multiplication and Division Properties of Equality<\/h2>\r\n<span style=\"color: #000000;\">Just as we can add or subtract the same exact quantity on both sides of an equation, we can also multiply or divide both sides of an equation by the same quantity to write an equivalent equation. This makes sense because multiplication is just repeated addition and division is multiplication by a reciprocal. <\/span>\r\n\r\n<span style=\"color: #000000;\">For example, to solve the equation [latex]3x=15[\/latex] we need to isolate the variable [latex]x[\/latex] which is multiplied by [latex]3[\/latex]. To \"undo\" this multiplication, we divide both sides of the equation by [latex]3[\/latex]:\u00a0 <\/span><span style=\"color: #000000;\">[latex]\\frac{3x}\\color{blue}{{3}}=\\frac{15}\\color{blue}{{3}}[\/latex]. <\/span><span style=\"color: #000000;\">Since [latex]\\frac{3}{3}=1[\/latex] and [latex]1x=x[\/latex], this simplifies to [latex]x=5[\/latex] and we have our solution.<\/span>\r\n\r\n<span style=\"color: #000000;\">On the other hand, in the equation [latex]\\frac{1}{2}x=-5[\/latex] the variable [latex]x[\/latex] is multiplied by [latex]\\frac{1}{2}[\/latex]. To isolate the variable we need to turn\u00a0[latex]\\frac{1}{2}[\/latex] into\u00a0[latex]1[\/latex] by multiplying by the reciprocal [latex]2[\/latex]: \u00a0 \u00a0 <\/span><span style=\"color: #000000;\">[latex]\\color{blue}{2}\\left(\\frac{1}{2}x\\right)=\\color{blue}{2}(-5)[\/latex]. This simplifies to [latex]x=-10[\/latex] and we have our solution.<\/span>\r\n\r\n<span style=\"color: #000000;\">This characteristic of equations is generalized in the <strong>M<\/strong><b>ultiplication &amp; Division Property of Equality<\/b>.<\/span>\r\n<div class=\"textbox shaded\">\r\n<h3 class=\"title\"><span style=\"color: #000000;\">multiplication &amp; Division Property of Equality<\/span><\/h3>\r\n<span style=\"color: #000000;\">For all real numbers [latex]a,b,c[\/latex], and [latex]c\\ne 0[\/latex], if [latex]a=b[\/latex], then [latex]ac=bc[\/latex] and [latex]\\Large\\frac{a}{c}\\normalsize =\\Large\\frac{b}{c}[\/latex].<\/span>\r\n\r\n<span style=\"color: #000000;\">Multiplying or dividing the same non-zero* term to both sides of an equation will result in an <strong><em>equivalent equation<\/em><\/strong>.<\/span>\r\n\r\n<\/div>\r\n<span style=\"color: #000000;\">*Technically we could multiply both sides of an equation by zero but that would wipe out our entire equation to [latex]0=0[\/latex]. However, we can never divide by zero as that is undefined. <\/span>\r\n\r\n<span style=\"color: #000000;\">When the equation involves multiplication or division, we can \u201cundo\u201d these operations by using the inverse operation to isolate the variable.<\/span>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3><span style=\"color: #000000;\">Example<\/span><\/h3>\r\n<span style=\"color: #000000;\">Solve [latex]3x=24[\/latex]. When you are done, check your solution.<\/span>\r\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\r\n<span style=\"color: #000000;\">Divide both sides of the equation by [latex]3[\/latex] to isolate the variable (this is will give you a coefficient of \u00a0[latex]1[\/latex]).\u00a0Dividing by [latex]3[\/latex] is the same as multiplying by [latex] \\frac{1}{3}[\/latex].<\/span>\r\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex]\\begin{array}{r}\\underline{3x}=\\underline{24}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\\\x=8\\,\\,\\,\\end{array}[\/latex]<\/span><\/p>\r\n<span style=\"color: #000000;\">Check by substituting your solution, [latex]8[\/latex], for the variable in the original equation.<\/span>\r\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex]\\begin{array}{r}3x=24 \\\\ 3\\cdot8=24 \\\\ 24=24\\end{array}[\/latex]<\/span><\/p>\r\n<span style=\"color: #000000;\">The solution is correct!<\/span>\r\n<h4>Answer<\/h4>\r\n<span style=\"color: #000000;\">[latex]x=8[\/latex]<\/span>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3><span style=\"color: #000000;\">example<\/span><\/h3>\r\n<span style=\"color: #000000;\">Solve: [latex]\\frac{a}{-7} =-42[\/latex]<\/span>\r\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\r\n<span style=\"color: #000000;\">Here [latex]a[\/latex] is divided by [latex]-7[\/latex]. We can multiply both sides by [latex]-7[\/latex] to isolate [latex]a[\/latex].<\/span>\r\n<table id=\"eip-id1168468288515\" class=\"unnumbered unstyled\" summary=\"The top shows a over negative 7 equals negative 42. The next line says \">\r\n<tbody>\r\n<tr>\r\n<th colspan=\"2\"><span style=\"color: #000000;\">[latex]\\Large\\frac{a}{-7}\\normalsize =-42[\/latex]<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td><span style=\"color: #000000;\">[latex]\\Large\\frac{a}{-7}\\normalsize =-42[\/latex]<\/span><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><span style=\"color: #000000;\">Multiply both sides by [latex]-7[\/latex] .<\/span><\/td>\r\n<td><span style=\"color: #000000;\">[latex]\\color{red}{-7}(\\frac{a}{-7})=\\color{red}{-7}(-42)[\/latex]<\/span>\r\n\r\n<span style=\"color: #000000;\">[latex]<\/span>\r\n\r\n<span style=\"color: #000000;\">\\frac{-7a}{-7}=294[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><span style=\"color: #000000;\">Simplify.<\/span><\/td>\r\n<td><span style=\"color: #000000;\">[latex]a=294[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><span style=\"color: #000000;\">Check your answer.<\/span><\/td>\r\n<td><span style=\"color: #000000;\">[latex]\\frac{a}{-7}=-42[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><span style=\"color: #000000;\">Let [latex]a=294[\/latex] .<\/span><\/td>\r\n<td><span style=\"color: #000000;\">[latex]\\frac{\\color{red}{294}}{-7}\\stackrel{\\text{?}}{=}-42[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><span style=\"color: #000000;\">[latex]-42=-42\\quad\\checkmark[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try\u00a0it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141868&amp;theme=oea&amp;iframe_resize_id=mom21[\/embed]\r\n\r\n<\/div>\r\n<span style=\"color: #000000;\">Another way to think about solving an equation when the operation is multiplication or division is that we want to multiply the coefficient by the multiplicative inverse (reciprocal) in order to change the coefficient to [latex]1[\/latex].<\/span>\r\n\r\n<span style=\"color: #000000;\">In the following example, we\u00a0change the coefficient to \u00a0[latex]1[\/latex] by multiplying by the multiplicative inverse of [latex]\\frac{1}{2}[\/latex].<\/span>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3><span style=\"color: #000000;\">Example<\/span><\/h3>\r\n<span style=\"color: #000000;\">Solve [latex] \\frac{1}{2}x=8[\/latex] for [latex]x[\/latex].<\/span>\r\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\r\n<span style=\"color: #000000;\">We can multiply both sides by the reciprocal of [latex]\\frac{1}{2}[\/latex], which is [latex]\\frac{2}{1}[\/latex].<\/span>\r\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex]\\begin{array}{r}\\color{blue}{\\left(\\frac{2}{1}\\right)}\\frac{1}{2 }{ x }=\\color{blue}{\\left(\\frac{2}{1}\\right)}8\\\\(1)x= 16\\,\\,\\,\\,\\\\{ x }=16\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/span><\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<span style=\"color: #000000;\">The video shows examples of how to use the Multiplication and Division Property of Equality to solve one-step equations with integers and fractions.<\/span>\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=BN7iVWWl2y0&amp;feature=youtu.be\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-4.docx\">Transcript-6.1.1-4<\/a>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n<span style=\"color: #000000;\">Solve: [latex]4x=-28[\/latex]<\/span>\r\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\r\n<span style=\"color: #000000;\">To solve this equation, we use the Division Property of Equality to divide both sides by [latex]4[\/latex].<\/span>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th colspan=\"2\">Solving [latex]4x=-28[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4x=-28[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide both sides by 4 to undo the multiplication.<\/td>\r\n<td>[latex]\\Large\\frac{4x}{\\color{red}4}\\normalsize =\\Large\\frac{-28}{\\color{red}4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x =-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check your answer.<\/td>\r\n<td>[latex]4x=-28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Let [latex]x=-7[\/latex]. Substitute [latex]-7[\/latex] for x.<\/td>\r\n<td>[latex]4(\\color{red}{-7})\\stackrel{\\text{?}}{=}-28[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>\u00a0[latex]-28=-28[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span style=\"color: #000000;\">Since this is a true statement, [latex]x=-7[\/latex] is a solution to [latex]4x=-28[\/latex].<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try\u00a0it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141857&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<span style=\"color: #000000;\">As we solve equations that require several steps, it is not unusual to end up with an equation that looks like the one in the next example, with a negative sign in front of the variable.\u00a0\u00a0<\/span>\r\n<div class=\"textbox exercises\">\r\n<h3><span style=\"color: #000000;\">example<\/span><\/h3>\r\n<span style=\"color: #000000;\">Solve: [latex]-r=2[\/latex]<\/span>\r\n\r\n&nbsp;\r\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\r\n<span style=\"color: #000000;\">Remember [latex]-r[\/latex] is equivalent to [latex]-1r[\/latex].<\/span>\r\n<table id=\"eip-id1168469604717\" class=\"unnumbered unstyled\" summary=\"The first line says negative r equals 2. The next line says \">\r\n<tbody>\r\n<tr>\r\n<th colspan=\"2\">Solving [latex]-r=2[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-r=2[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite [latex]-r[\/latex] as [latex]-1r[\/latex] .<\/td>\r\n<td>[latex]-1r=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide both sides by [latex]-1[\/latex] .<\/td>\r\n<td>[latex]\\Large\\frac{-1r}{\\color{red}{-1}}\\normalsize =\\Large\\frac{2}{\\color{red}{-1}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]r=-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.<\/td>\r\n<td>[latex]-r=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]r=-2[\/latex]<\/td>\r\n<td>[latex]-(\\color{red}{-2})\\stackrel{\\text{?}}{=}2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]2=2\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try\u00a0it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141865&amp;theme=oea&amp;iframe_resize_id=mom22[\/embed]\r\n\r\n<\/div>\r\nThe equations [latex]x=4[\/latex] and\u00a0[latex]4=x[\/latex] are equivalent. Both say that\u00a0[latex]x[\/latex] is equal to\u00a0[latex]4[\/latex]. This is an example of <em><strong>The Reflection Property of Equality<\/strong><\/em>.\r\n<div class=\"textbox shaded\">\r\n<h3>The Reflection Property of Equality<\/h3>\r\n<p style=\"text-align: center;\">If [latex]a=b[\/latex], then [latex]b=a[\/latex], for all real numbers [latex]a[\/latex] and\u00a0[latex]b[\/latex].<\/p>\r\n\r\n<\/div>\r\nThis implies that it does not matter which side of the equation the variable term ends up on.\r\n\r\n<span style=\"color: #000000;\">The next video includes examples of using the division and multiplication properties to solve equations with the variable on the right side of the equal sign.<\/span>\r\n\r\nhttps:\/\/youtu.be\/TB1rkPbF8rA\r\n<h2><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-5.docx\">Transcript-6.1.1-5<\/a><\/h2>\r\n<h2>Two-Step Linear Equations<\/h2>\r\n<span style=\"color: #000000;\">If the equation is in the form [latex]ax+b=c[\/latex], where [latex]x[\/latex] is the variable, we can solve the equation as before. First we must isolate the [latex]x-[\/latex]term by \u201cundoing\u201d the addition or subtraction. Then we isolate the variable by \u201cundoing\u201d the multiplication or division.<\/span>\r\n<div class=\"textbox exercises\">\r\n<h3><span style=\"color: #000000;\">Example<\/span><\/h3>\r\n<span style=\"color: #000000;\">1. Solve: [latex]4x+6=-14[\/latex]<\/span>\r\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\r\n<span style=\"color: #000000;\">In this equation, the variable is only on the left side. It makes sense to call the left side the variable side. Therefore, the right side will be the constant side.<\/span>\r\n<table style=\"width: 70%;\" summary=\"The top line says 4x plus 6 equals negative 14.\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 1179.02px;\" colspan=\"3\"><span style=\"color: #000000;\">Solving [latex]4x+6=-14[\/latex]<\/span><span style=\"color: #000000;\">\r\n<\/span><\/th>\r\n<\/tr>\r\n<tr style=\"height: 45.8594px;\">\r\n<td style=\"height: 45.8594px;\" colspan=\"2\"><span style=\"color: #000000;\">Since the left side is the variable side, the 6 is out of place. We must \"undo\" adding [latex]6[\/latex] by subtracting [latex]6[\/latex], and to keep the equality we must subtract [latex]6[\/latex] from both sides. Use the Subtraction Property of Equality.<\/span><\/td>\r\n<td style=\"height: 45.8594px; width: 176px;\"><span style=\"color: #000000;\">[latex]4x+6\\color{red}{-6}=-14\\color{red}{-6}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Simplify.<\/span><\/td>\r\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]4x=-20[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Now all the [latex]x[\/latex] s are on the left and the constant on the right.<\/span><\/td>\r\n<td style=\"height: 15px; width: 176px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 60px;\">\r\n<td style=\"height: 60px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Use the Division Property of Equality.<\/span><\/td>\r\n<td style=\"height: 60px; width: 176px;\"><span style=\"color: #000000;\">[latex]\\Large\\frac{4x}{\\color{red}{4}}\\normalsize =\\Large\\frac{-20}{\\color{red}{4}}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Simplify.<\/span><\/td>\r\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]x=-5[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Check:<\/span><\/td>\r\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]4x+6=-14[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Let [latex]x=-5[\/latex] .<\/span><\/td>\r\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]4(\\color{red}{-5})+6=-14[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><\/td>\r\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]-20+6=-14[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><\/td>\r\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]-14=-14\\quad\\checkmark[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<span style=\"color: #000000;\">2. Solve: [latex]2y - 7=15[\/latex]<\/span>\r\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\r\n<span style=\"color: #000000;\">Notice that the variable is only on the left side of the equation, so this will be the variable side, and the right side will be the constant side. Since the left side is the variable side, the [latex]7[\/latex] is out of place. It is subtracted from the [latex]2y[\/latex], so to \"undo\" subtraction, add [latex]7[\/latex] to both sides.<\/span>\r\n<table id=\"eip-id1168469592645\" class=\"unnumbered unstyled\" summary=\"The first line says 2y minus 7 equals 15. The left side is labeled \">\r\n<tbody>\r\n<tr>\r\n<th colspan=\"3\">Solving <span style=\"color: #000000;\">[latex]2y - 7=15[\/latex]<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2y-7[\/latex] is the side containing a <span style=\"color: #000000;\"><span style=\"color: #ff0000;\">variable<\/span>.<\/span><span style=\"color: #000000;\">[latex]15[\/latex] is the side containing only a <span style=\"color: #ff0000;\">constant<\/span>.<\/span><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Add [latex]7[\/latex] to both sides.<\/td>\r\n<td>[latex]2y-7\\color{red}{+7}=15\\color{red}{+7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]2y=22[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"3\">The variables are now on one side and the constants on the other.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Divide both sides by [latex]2[\/latex].<\/td>\r\n<td>[latex]\\Large\\frac{2y}{\\color{red}{2}}\\normalsize =\\Large\\frac{22}{\\color{red}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]y=11[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Check:<\/td>\r\n<td>[latex]2y-7=15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Let [latex]y=11[\/latex] .<\/td>\r\n<td>[latex]2\\cdot\\color{red}{11}-7\\stackrel{\\text{?}}{=}15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>[latex]22-7\\stackrel{\\text{?}}{=}15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>[latex]15=15\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\n<span style=\"color: #000000;\">Solve [latex]3y+2=11[\/latex]<\/span>\r\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\r\n<span style=\"color: #000000;\">Subtract [latex]2[\/latex] from both sides of the equation to get the term with the variable by itself.<\/span>\r\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex] \\displaystyle \\begin{array}{r}3y+2\\,\\,\\,=\\,\\,11\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\color{blue}{-2}\\,\\,\\,\\,\\,\\,\\,\\,\\color{blue}{-2}}\\\\3y\\,\\,\\,\\,=\\,\\,\\,\\,\\,9\\end{array}[\/latex]<\/span><\/p>\r\n<span style=\"color: #000000;\">Divide both sides of the equation by [latex]3[\/latex] to get a coefficient of [latex]1[\/latex] for the variable.<\/span>\r\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\underline{3y}\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\underline{9}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,9\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\,\\,\\,\\,=\\,\\,\\,\\,3\\end{array}[\/latex]<\/span><\/p>\r\n\r\n<h4>Answer<\/h4>\r\n<span style=\"color: #000000;\">[latex]y=3[\/latex]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try\u00a0It<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142131&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n<\/div>\r\n<span style=\"color: #000000;\">In the following video, we show examples of solving two step linear equations.<\/span>\r\n\r\nhttps:\/\/youtu.be\/fCyxSVQKeRw\r\n\r\n<a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-6.docx\">Transcript-6.1.1-6<\/a>\r\n\r\n<span style=\"color: #000000;\">Remember to check the solution of an algebraic equation by substituting the value of the variable into the original equation.<\/span>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>","rendered":"<nav style=\"background-color: #007fab;\" role=\"navigation\">\n<div class=\"book-title-wrapper\">\n<div class=\"bombadil-logo\"><a href=\"https:\/\/courses.lumenlearning.com\/\"><img decoding=\"async\" src=\"https:\/\/courses.lumenlearning.com\/boundless-algebra\/wp-content\/themes\/bombadil\/assets\/images\/LumenOnDark-150x69.png\" alt=\"Lumen\" \/><\/a><\/div>\n<\/div>\n<\/nav>\n<div id=\"content\" role=\"main\">\n<h1 class=\"entry-title\">Solving Linear Equations<\/h1>\n<div id=\"post-902\" class=\"standard post-902 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div id=\"wpipa-1404-container\" class=\"wpipa-container wpipa-align-center\" data-id=\"1404\" data-variation=\"none\">\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Determine whether or not an equation in one variable is defined as a linear equation<\/li>\n<li>Use the addition and subtraction property to solve a linear equation<\/li>\n<li>Use the multiplication and division property to solve a linear equation<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h1>KEY WORDS<\/h1>\n<ul>\n<li><strong>Equality<\/strong>: The state of two or more entities having the same value.<\/li>\n<li><strong>Equation<\/strong>: a statement that two expressions are equal.<\/li>\n<li><strong>Solution<\/strong>: a number or numbers that make an equation true.<\/li>\n<li><strong>Equivalent equations:<\/strong> two or more equations that have identical solutions.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h2>Linear Equations<\/h2>\n<p>An equation with just <strong>one variable<\/strong> is said to be\u00a0<em><strong>linear<\/strong><\/em> when the highest power on the variable is [latex]1[\/latex]. Remember that [latex]{x}^{1}[\/latex] is equivalent to [latex]x[\/latex], so any equation that can be simplified to [latex]ax+b=c[\/latex] (where [latex]a,b,c[\/latex] are real numbers) is a <strong>linear equation in one variable<\/strong>. For example, the equation [latex]2x+3=7, \\;5x=7,\\; -\\frac{3}{4}x+\\frac{7}{3}=\\frac{6}{5}[\/latex] are all linear equations because the highest power of [latex]x[\/latex] is [latex]1[\/latex]. On the other hand, the equation [latex]{x}^3+2{x}^2=4x-2[\/latex] is NOT linear because the highest power of\u00a0[latex]x[\/latex] is\u00a0[latex]3[\/latex], not\u00a0[latex]1[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Determine whether the equation is linear:<\/p>\n<ul>\n<li>a)\u00a0[latex]3x-7=6x+2[\/latex]<\/li>\n<li>b)\u00a0[latex]{x}^{3}-5{x}^{2}=5x+6[\/latex]<\/li>\n<li>c)\u00a0[latex]x(x+6)=8[\/latex]<\/li>\n<li>d)\u00a0[latex]x-\\frac{3}{8} =\\frac{1}{2}[\/latex]<\/li>\n<\/ul>\n<h4><strong>Solution:<\/strong><\/h4>\n<table id=\"eip-id1168466426761\" class=\"unnumbered unstyled\" summary=\"The top line says 4y plus 3 equals 8y. Beside this is\">\n<tbody>\n<tr>\n<th>Equation<\/th>\n<th>Explanation<\/th>\n<\/tr>\n<tr>\n<td>a) [latex]3x-7=6x+2[\/latex]<\/td>\n<td>Yes, it is linear since the highest power of [latex]x[\/latex] is\u00a0[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>b) [latex]{x}^{3}-5{x}^{2}=5x+6[\/latex]<\/td>\n<td>No, it is NOT linear since the highest power of\u00a0[latex]x[\/latex] is\u00a0[latex]3[\/latex], not\u00a0[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>c) [latex]x(x+6)=8[\/latex]<\/td>\n<td>Use the distributive property to simplify the left side of the equation:\u00a0[latex]{x}^{2}+6x=8[\/latex]. Now we can see that the equation is NOT linear\u00a0since the highest power of\u00a0[latex]x[\/latex] is\u00a0[latex]2[\/latex], not\u00a0[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>d) [latex]x-\\frac{3}{8} =\\frac{1}{2}[\/latex]<\/td>\n<td>Yes, it is linear since the highest power of [latex]x[\/latex] is\u00a0[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p>Determine whether the equation [latex]6x-7=8{x}^{2}+5[\/latex] is linear.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH411001\">Show Solution<\/span><\/p>\n<div id=\"qH411001\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<p>[latex]6x-7=8{x}^{2}+5[\/latex] is NOT linear since the highest power of [latex]x[\/latex] is [latex]2[\/latex], not [latex]1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p>Determine whether the equation [latex]\\frac{4}{5}x+6=\\frac{-7}{3}x[\/latex] is linear.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qH411002\">Show Solution<\/span><\/p>\n<div id=\"qH411002\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<p>[latex]\\frac{4}{5}x+6=\\frac{-7}{3}x[\/latex] is linear since the highest power of [latex]x[\/latex] is [latex]1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>The Addition and Subtraction Property of Equality<\/h2>\n<p>When an equation contains a variable such as [latex]x[\/latex], this variable is considered an <strong><em>unknown<\/em><\/strong> value. In many cases, we can find the possible values for [latex]x[\/latex] that would make the equation true. We can\u00a0<em><strong>solve\u00a0<\/strong><\/em>the equation for [latex]x[\/latex].<\/p>\n<p>For some equations like [latex]x + 3 =5[\/latex] it is easy to guess the solution: the only possible value of [latex]x[\/latex] is [latex]2[\/latex], because [latex]2 + 3 = 5[\/latex].\u00a0However, it becomes useful to have a process for finding solutions for unknowns as problems become more complex.<\/p>\n<p>In order to solve an equation, we need to\u00a0<em><b style=\"color: #000000; font-size: 1rem; orphans: 1; text-align: initial;\">isolate the variable<\/b><\/em><span style=\"color: #000000; font-size: 1rem; font-weight: normal; orphans: 1; text-align: initial;\">. Isolating the variable means rewriting an <em><strong>equivalent equation<\/strong><\/em> in which the variable is on one side of the equation and everything else is on the other side of the equation. The variable is an <em><strong>unknown<\/strong><\/em> quantity whose value we are trying to find.\u00a0We have a solution when we reorganize the equation into the form [latex]x[\/latex] = a constant.<\/span><\/p>\n<\/div>\n<div class=\"entry-content\"><span style=\"color: #000000; font-size: 1rem; orphans: 1; text-align: initial;\">An equation is two expressions that are set equal to each other. If we perform an operation (+, \u2013, \u00b7, \u00f7, etc) to one side of the equation, it will upset the balance of the equation. Our goal is to keep the balance of the equation intact. This can be likened to a balance scale where the equation [latex]x+3=5[\/latex] is illustrated:<\/span><\/div>\n<div><\/div>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1853 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/09\/26203113\/Scale-equation-300x282.png\" alt=\"Scale\" width=\"300\" height=\"282\" \/><\/p>\n<p>The second we add or subtract a &#8220;weight&#8221; to one side of the scale, it will become unbalanced. In order to right that balance, we must add or subtract the same &#8220;weight&#8221; to the other side of the sale.<\/p>\n<p>To solve this equation for [latex]x[\/latex], we need to isolate [latex]x[\/latex] by removing the [latex]+3[\/latex] from the left side of the equation. We do this this by &#8220;undoing&#8221; the addition of\u00a0[latex]+3[\/latex] by subtracting\u00a0[latex]3[\/latex].\u00a0 In other words we use the inverse operation of addition, which is subtraction. But if we subtract\u00a0[latex]3[\/latex] from the left side of the equation, we must also subtract\u00a0[latex]3[\/latex] from the right side to keep the equation balanced.<\/p>\n<p style=\"text-align: center;\">[latex]x+3\\;\\;\\;\\;\\;\\;=5 \\\\ x+3\\color{blue}{-3}=5\\color{blue}{-3} \\\\ x\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\,=2[\/latex]<\/p>\n<p>We can verify that [latex]x=2[\/latex] is indeed the solution by substituting\u00a0[latex]2[\/latex] in for\u00a0[latex]x[\/latex] in the original equation: [latex]x+3=5 \\\\ 2+3=5 \\\\ 5=5[\/latex] Since this is true,\u00a0[latex]x=2[\/latex] is the solution of\u00a0[latex]x+3=5[\/latex].<\/p>\n<p><span style=\"color: #000000;\">If the equation is [latex]x-5=9[\/latex], we isolate the variable by adding [latex]5[\/latex] to both sides of the equation. This is because, to isolate [latex]x[\/latex] we must &#8220;undo&#8221; subtract [latex]5[\/latex] by adding [latex]5[\/latex]:\u00a0<\/span><\/p>\n<p>[latex]x-5=9 \\\\ x-5\\color{blue}{+5}=9\\color{blue}{+5} \\\\ x\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\,=14[\/latex]<\/p>\n<p>Again, we can verify the solution:\u00a0[latex]x-5\\;\\;\\;\\;\\;\\;=9 \\\\ 14 - 5 = 9 \\\\ 9=9[\/latex] True.<\/p>\n<p>When we add or subtract the same term to both sides of an equation, we get an <em><strong>equivalent equation<\/strong><\/em>. <strong>Equivalent equations<\/strong> are two or more equations that have identical solutions. This leads us to the the <em><strong>Addition and Subtraction Property of Equality<\/strong><\/em>.<\/p>\n<div class=\"textbox shaded\">\n<h3 id=\"fs-id1166490863495\"><strong>Addition &amp; subtraction Property of Equality<\/strong><\/h3>\n<p><span style=\"color: #000000;\">For all real numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex] and [latex]a-c=b-c[\/latex].<\/span><\/p>\n<p><span style=\"color: #000000;\">Adding or subtracting the same term to both sides of an equation will result in an <strong><em>equivalent equation<\/em><\/strong>.<\/span><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p><span style=\"color: #000000;\">Solve: [latex]x+11=-3[\/latex]<\/span><\/p>\n<h4><span style=\"color: #000000;\">Solution:<\/span><\/h4>\n<p><span style=\"color: #000000;\">To isolate [latex]x[\/latex], we undo the addition of [latex]11[\/latex] by using the Subtraction Property of Equality.<\/span><\/p>\n<table id=\"eip-id1168468291100\" class=\"unnumbered unstyled\" style=\"height: 90px;\" summary=\"The top line says x plus 11 equals negative 3. The next line says,\">\n<tbody>\n<tr>\n<th style=\"width: 349.656px;\" colspan=\"3\">Solving [latex]x+11=-3[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"height: 11px;\" colspan=\"2\"><\/td>\n<td style=\"height: 11px; width: 206.656px;\">[latex]x+11=-3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"height: 23px; width: 349.656px;\" colspan=\"2\">Subtract 11 from each side to &#8220;undo&#8221; the addition.<\/td>\n<td style=\"height: 23px; width: 206.656px;\">[latex]x+11\\color{red}{-11}=-3\\color{red}{-11}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"height: 11px; width: 349.656px;\" colspan=\"2\">Simplify.<\/td>\n<td style=\"height: 11px; width: 206.656px;\">[latex]x=-14[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"height: 11px; width: 121.656px;\">Check:<\/td>\n<td style=\"height: 11px; width: 216.656px;\">[latex]x+11=-3[\/latex]<\/td>\n<td style=\"height: 11px; width: 206.656px;\"><\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"height: 23px; width: 121.656px;\">Substitute [latex]x=-14[\/latex] .<\/td>\n<td style=\"height: 23px; width: 216.656px;\">[latex]\\color{red}{-14}+11\\stackrel{\\text{?}}{=}-3[\/latex]<\/td>\n<td style=\"height: 23px; width: 206.656px;\"><\/td>\n<\/tr>\n<tr style=\"height: 11px;\">\n<td style=\"height: 11px; width: 121.656px;\"><\/td>\n<td style=\"height: 11px; width: 216.656px;\">[latex]-3=-3\\quad\\checkmark[\/latex]<\/td>\n<td style=\"height: 11px; width: 206.656px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"color: #000000;\">Since [latex]x=-14[\/latex] makes [latex]x+11=-3[\/latex] a true statement, we know that it is a solution to the equation.<\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141721\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141721&#38;theme=oea&#38;iframe_resize_id=ohm141721&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p><span style=\"color: #000000;\">Solve: [latex]m - 4=-5[\/latex]<\/span><\/p>\n<h4><span style=\"color: #000000;\">Solution:<\/span><\/h4>\n<table id=\"eip-id1168467157298\" class=\"unnumbered unstyled\" summary=\"The first line says m minus 4 equals negative 5. The next line says,\">\n<tbody>\n<tr>\n<th colspan=\"3\">Solving [latex]m-4=-5[\/latex]<\/th>\n<\/tr>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>[latex]m-4=-5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Add 4 to each side to &#8220;undo&#8221; the subtraction.<\/td>\n<td>[latex]m-4\\color{red}{+4}=-5\\color{red}{+4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]m=-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]m-4=-5[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]m=-1[\/latex] .<\/td>\n<td>[latex]\\color{red}{-1}+4\\stackrel{\\text{?}}{=}-5[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-5=-5\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td>The solution to [latex]m - 4=-5[\/latex] is [latex]m=-1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141723\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141723&#38;theme=oea&#38;iframe_resize_id=ohm141723&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Solve One Step Equations By Add and Subtract Whole Numbers (Variable on Left)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/yqdlj0lv7Cc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"post-902\" class=\"standard post-902 chapter type-chapter status-publish hentry\"><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-1.docx\">Transcript-6.1.1-1<\/a><\/div>\n<h2>Linear Equations Containing Fractions<\/h2>\n<p><span style=\"color: #000000;\">It is not uncommon to encounter equations that contain fractions; therefore, in the following examples, we will demonstrate how to use the addition property of equality to solve an equation with fractions.<\/span><\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p><span style=\"color: #000000;\">Solve: [latex]n-\\frac{3}{8} =\\frac{1}{2}[\/latex]<\/span><\/p>\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\n<table id=\"eip-id1168468255179\" class=\"unnumbered unstyled\" style=\"width: 602px;\" summary=\"The first line says n minus 3 eighths equals one half. The next line says,\">\n<tbody>\n<tr>\n<th style=\"width: 313.391px;\" colspan=\"3\">Solving [latex]n-\\frac{3}{8}=\\frac{1}{2}[\/latex]<\/th>\n<\/tr>\n<tr>\n<td colspan=\"2\"><\/td>\n<td style=\"width: 255.609px;\">[latex]n-\\frac{3}{8} =\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 313.391px;\" colspan=\"2\">Use the Addition Property of Equality.<\/td>\n<td style=\"width: 255.609px;\">[latex]n-\\frac{3}{8}\\color{red}{+\\frac{3}{8}} =\\frac{1}{2}\\color{red}{+\\frac{3}{8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 313.391px;\" colspan=\"2\">Find the LCD to add the fractions on the right.<\/td>\n<td style=\"width: 255.609px;\">[latex]n-\\frac{3}{8} +\\frac{3}{8} =\\frac{4}{8} +\\frac{3}{8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 313.391px;\" colspan=\"2\">Simplify.<\/td>\n<td style=\"width: 255.609px;\">[latex]n=\\frac{7}{8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 142px;\">Check:<\/td>\n<td style=\"width: 171.391px;\">[latex]n-\\frac{3}{8} =\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 255.609px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 142px;\">Substitute [latex]n=\\color{red}{\\frac{7}{8}}[\/latex]<\/td>\n<td style=\"width: 171.391px;\">[latex]\\color{red}{\\frac{7}{8}} -\\frac{3}{8}\\stackrel{\\text{?}}{=}\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 255.609px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 142px;\">Subtract.<\/td>\n<td style=\"width: 171.391px;\">[latex]\\frac{4}{8}\\stackrel{\\text{?}}{=}\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 255.609px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 142px;\">Simplify.<\/td>\n<td style=\"width: 171.391px;\">[latex]\\frac{1}{2}\\normalsize =\\frac{1}{2}\\normalsize\\quad\\checkmark[\/latex]<\/td>\n<td style=\"width: 255.609px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 142px;\">The solution checks.<\/td>\n<td style=\"width: 171.391px;\"><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3><span style=\"color: #000000;\">TRY<\/span>\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141732\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141732&#38;theme=oea&#38;iframe_resize_id=ohm141732&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #000000;\">Watch this video for more examples of solving equations that include fractions and require addition or subtraction.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Solve One Step Equations With Fraction by Adding or Subtracting\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KmOvCakGEgM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-2-1.docx\">Transcript-6.1.1-2<\/a><\/h2>\n<h2>Linear Equations Containing Decimals<\/h2>\n<p>Decimals will be encountered any time money or metric measurements are used.<\/p>\n<div class=\"textbox exercises\">\n<h3>eXAMPLE<\/h3>\n<p><span style=\"color: #000000;\">Solve [latex]a - 3.7=4.3[\/latex]<\/span><\/p>\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\n<table id=\"eip-id1168468751492\" class=\"unnumbered unstyled\" style=\"width: 730px;\" summary=\"The top line says a minus 3.7 equals 4.3. The next line says\">\n<tbody>\n<tr>\n<th style=\"width: 516.391px;\" colspan=\"3\">Solving [latex]a-3.7=4.3[\/latex]<\/th>\n<\/tr>\n<tr>\n<td colspan=\"2\"><\/td>\n<td style=\"width: 180.609px;\">[latex]a-3.7=4.3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 516.391px;\" colspan=\"2\">Use the Addition Property of Equality.<\/td>\n<td style=\"width: 180.609px;\">[latex]a-3.7\\color{red}{+3.7}=4.3\\color{red}{+3.7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 516.391px;\" colspan=\"2\">Add.<\/td>\n<td style=\"width: 180.609px;\">[latex]a=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 176px;\">Check:<\/td>\n<td style=\"width: 340.391px;\">[latex]a-3.7=4.3[\/latex]<\/td>\n<td style=\"width: 180.609px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 176px;\">Substitute [latex]a=8[\/latex] .<\/td>\n<td style=\"width: 340.391px;\">[latex]\\color{red}{8}-3.7\\stackrel{\\text{?}}{=}4.3[\/latex]<\/td>\n<td style=\"width: 180.609px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 176px;\">Simplify.<\/td>\n<td style=\"width: 340.391px;\">[latex]4.3=4.3\\quad\\checkmark[\/latex]<\/td>\n<td style=\"width: 180.609px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 176px;\">The solution checks.<\/td>\n<td style=\"width: 340.391px;\"><\/td>\n<td style=\"width: 180.609px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141736\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141736&#38;theme=oea&#38;iframe_resize_id=ohm141736&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Solve a One Step Equation With Decimals by Adding and Subtracting\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/a6YYJN_bHKs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-3.docx\">Transcript-6.1.1-3<\/a><\/h2>\n<h2 id=\"title2\">The Multiplication and Division Properties of Equality<\/h2>\n<p><span style=\"color: #000000;\">Just as we can add or subtract the same exact quantity on both sides of an equation, we can also multiply or divide both sides of an equation by the same quantity to write an equivalent equation. This makes sense because multiplication is just repeated addition and division is multiplication by a reciprocal. <\/span><\/p>\n<p><span style=\"color: #000000;\">For example, to solve the equation [latex]3x=15[\/latex] we need to isolate the variable [latex]x[\/latex] which is multiplied by [latex]3[\/latex]. To &#8220;undo&#8221; this multiplication, we divide both sides of the equation by [latex]3[\/latex]:\u00a0 <\/span><span style=\"color: #000000;\">[latex]\\frac{3x}\\color{blue}{{3}}=\\frac{15}\\color{blue}{{3}}[\/latex]. <\/span><span style=\"color: #000000;\">Since [latex]\\frac{3}{3}=1[\/latex] and [latex]1x=x[\/latex], this simplifies to [latex]x=5[\/latex] and we have our solution.<\/span><\/p>\n<p><span style=\"color: #000000;\">On the other hand, in the equation [latex]\\frac{1}{2}x=-5[\/latex] the variable [latex]x[\/latex] is multiplied by [latex]\\frac{1}{2}[\/latex]. To isolate the variable we need to turn\u00a0[latex]\\frac{1}{2}[\/latex] into\u00a0[latex]1[\/latex] by multiplying by the reciprocal [latex]2[\/latex]: \u00a0 \u00a0 <\/span><span style=\"color: #000000;\">[latex]\\color{blue}{2}\\left(\\frac{1}{2}x\\right)=\\color{blue}{2}(-5)[\/latex]. This simplifies to [latex]x=-10[\/latex] and we have our solution.<\/span><\/p>\n<p><span style=\"color: #000000;\">This characteristic of equations is generalized in the <strong>M<\/strong><b>ultiplication &amp; Division Property of Equality<\/b>.<\/span><\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\"><span style=\"color: #000000;\">multiplication &amp; Division Property of Equality<\/span><\/h3>\n<p><span style=\"color: #000000;\">For all real numbers [latex]a,b,c[\/latex], and [latex]c\\ne 0[\/latex], if [latex]a=b[\/latex], then [latex]ac=bc[\/latex] and [latex]\\Large\\frac{a}{c}\\normalsize =\\Large\\frac{b}{c}[\/latex].<\/span><\/p>\n<p><span style=\"color: #000000;\">Multiplying or dividing the same non-zero* term to both sides of an equation will result in an <strong><em>equivalent equation<\/em><\/strong>.<\/span><\/p>\n<\/div>\n<p><span style=\"color: #000000;\">*Technically we could multiply both sides of an equation by zero but that would wipe out our entire equation to [latex]0=0[\/latex]. However, we can never divide by zero as that is undefined. <\/span><\/p>\n<p><span style=\"color: #000000;\">When the equation involves multiplication or division, we can \u201cundo\u201d these operations by using the inverse operation to isolate the variable.<\/span><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3><span style=\"color: #000000;\">Example<\/span><\/h3>\n<p><span style=\"color: #000000;\">Solve [latex]3x=24[\/latex]. When you are done, check your solution.<\/span><\/p>\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\n<p><span style=\"color: #000000;\">Divide both sides of the equation by [latex]3[\/latex] to isolate the variable (this is will give you a coefficient of \u00a0[latex]1[\/latex]).\u00a0Dividing by [latex]3[\/latex] is the same as multiplying by [latex]\\frac{1}{3}[\/latex].<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex]\\begin{array}{r}\\underline{3x}=\\underline{24}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\\\x=8\\,\\,\\,\\end{array}[\/latex]<\/span><\/p>\n<p><span style=\"color: #000000;\">Check by substituting your solution, [latex]8[\/latex], for the variable in the original equation.<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex]\\begin{array}{r}3x=24 \\\\ 3\\cdot8=24 \\\\ 24=24\\end{array}[\/latex]<\/span><\/p>\n<p><span style=\"color: #000000;\">The solution is correct!<\/span><\/p>\n<h4>Answer<\/h4>\n<p><span style=\"color: #000000;\">[latex]x=8[\/latex]<\/span><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3><span style=\"color: #000000;\">example<\/span><\/h3>\n<p><span style=\"color: #000000;\">Solve: [latex]\\frac{a}{-7} =-42[\/latex]<\/span><\/p>\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\n<p><span style=\"color: #000000;\">Here [latex]a[\/latex] is divided by [latex]-7[\/latex]. We can multiply both sides by [latex]-7[\/latex] to isolate [latex]a[\/latex].<\/span><\/p>\n<table id=\"eip-id1168468288515\" class=\"unnumbered unstyled\" summary=\"The top shows a over negative 7 equals negative 42. The next line says\">\n<tbody>\n<tr>\n<th colspan=\"2\"><span style=\"color: #000000;\">[latex]\\Large\\frac{a}{-7}\\normalsize =-42[\/latex]<\/span><\/th>\n<\/tr>\n<tr>\n<td><span style=\"color: #000000;\">[latex]\\Large\\frac{a}{-7}\\normalsize =-42[\/latex]<\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span style=\"color: #000000;\">Multiply both sides by [latex]-7[\/latex] .<\/span><\/td>\n<td><span style=\"color: #000000;\">[latex]\\color{red}{-7}(\\frac{a}{-7})=\\color{red}{-7}(-42)[\/latex]<\/span><\/p>\n<p><span style=\"color: #000000;\">[latex]<\/span>    <span style=\"color: #000000;\">\\frac{-7a}{-7}=294[\/latex]<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"color: #000000;\">Simplify.<\/span><\/td>\n<td><span style=\"color: #000000;\">[latex]a=294[\/latex]<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"color: #000000;\">Check your answer.<\/span><\/td>\n<td><span style=\"color: #000000;\">[latex]\\frac{a}{-7}=-42[\/latex]<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"color: #000000;\">Let [latex]a=294[\/latex] .<\/span><\/td>\n<td><span style=\"color: #000000;\">[latex]\\frac{\\color{red}{294}}{-7}\\stackrel{\\text{?}}{=}-42[\/latex]<\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span style=\"color: #000000;\">[latex]-42=-42\\quad\\checkmark[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141868\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141868&#38;theme=oea&#38;iframe_resize_id=ohm141868&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #000000;\">Another way to think about solving an equation when the operation is multiplication or division is that we want to multiply the coefficient by the multiplicative inverse (reciprocal) in order to change the coefficient to [latex]1[\/latex].<\/span><\/p>\n<p><span style=\"color: #000000;\">In the following example, we\u00a0change the coefficient to \u00a0[latex]1[\/latex] by multiplying by the multiplicative inverse of [latex]\\frac{1}{2}[\/latex].<\/span><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3><span style=\"color: #000000;\">Example<\/span><\/h3>\n<p><span style=\"color: #000000;\">Solve [latex]\\frac{1}{2}x=8[\/latex] for [latex]x[\/latex].<\/span><\/p>\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\n<p><span style=\"color: #000000;\">We can multiply both sides by the reciprocal of [latex]\\frac{1}{2}[\/latex], which is [latex]\\frac{2}{1}[\/latex].<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex]\\begin{array}{r}\\color{blue}{\\left(\\frac{2}{1}\\right)}\\frac{1}{2 }{ x }=\\color{blue}{\\left(\\frac{2}{1}\\right)}8\\\\(1)x= 16\\,\\,\\,\\,\\\\{ x }=16\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/span><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #000000;\">The video shows examples of how to use the Multiplication and Division Property of Equality to solve one-step equations with integers and fractions.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Solving One Step Equations Using Multiplication and Division (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BN7iVWWl2y0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-4.docx\">Transcript-6.1.1-4<\/a><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p><span style=\"color: #000000;\">Solve: [latex]4x=-28[\/latex]<\/span><\/p>\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\n<p><span style=\"color: #000000;\">To solve this equation, we use the Division Property of Equality to divide both sides by [latex]4[\/latex].<\/span><\/p>\n<table>\n<tbody>\n<tr>\n<th colspan=\"2\">Solving [latex]4x=-28[\/latex]<\/th>\n<\/tr>\n<tr>\n<td>[latex]4x=-28[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Divide both sides by 4 to undo the multiplication.<\/td>\n<td>[latex]\\Large\\frac{4x}{\\color{red}4}\\normalsize =\\Large\\frac{-28}{\\color{red}4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x =-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check your answer.<\/td>\n<td>[latex]4x=-28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Let [latex]x=-7[\/latex]. Substitute [latex]-7[\/latex] for x.<\/td>\n<td>[latex]4(\\color{red}{-7})\\stackrel{\\text{?}}{=}-28[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\u00a0[latex]-28=-28[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"color: #000000;\">Since this is a true statement, [latex]x=-7[\/latex] is a solution to [latex]4x=-28[\/latex].<\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141857\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141857&#38;theme=oea&#38;iframe_resize_id=ohm141857&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #000000;\">As we solve equations that require several steps, it is not unusual to end up with an equation that looks like the one in the next example, with a negative sign in front of the variable.\u00a0\u00a0<\/span><\/p>\n<div class=\"textbox exercises\">\n<h3><span style=\"color: #000000;\">example<\/span><\/h3>\n<p><span style=\"color: #000000;\">Solve: [latex]-r=2[\/latex]<\/span><\/p>\n<p>&nbsp;<\/p>\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\n<p><span style=\"color: #000000;\">Remember [latex]-r[\/latex] is equivalent to [latex]-1r[\/latex].<\/span><\/p>\n<table id=\"eip-id1168469604717\" class=\"unnumbered unstyled\" summary=\"The first line says negative r equals 2. The next line says\">\n<tbody>\n<tr>\n<th colspan=\"2\">Solving [latex]-r=2[\/latex]<\/th>\n<\/tr>\n<tr>\n<td>[latex]-r=2[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Rewrite [latex]-r[\/latex] as [latex]-1r[\/latex] .<\/td>\n<td>[latex]-1r=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide both sides by [latex]-1[\/latex] .<\/td>\n<td>[latex]\\Large\\frac{-1r}{\\color{red}{-1}}\\normalsize =\\Large\\frac{2}{\\color{red}{-1}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]r=-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.<\/td>\n<td>[latex]-r=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]r=-2[\/latex]<\/td>\n<td>[latex]-(\\color{red}{-2})\\stackrel{\\text{?}}{=}2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]2=2\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try\u00a0it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141865\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141865&#38;theme=oea&#38;iframe_resize_id=ohm141865&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The equations [latex]x=4[\/latex] and\u00a0[latex]4=x[\/latex] are equivalent. Both say that\u00a0[latex]x[\/latex] is equal to\u00a0[latex]4[\/latex]. This is an example of <em><strong>The Reflection Property of Equality<\/strong><\/em>.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Reflection Property of Equality<\/h3>\n<p style=\"text-align: center;\">If [latex]a=b[\/latex], then [latex]b=a[\/latex], for all real numbers [latex]a[\/latex] and\u00a0[latex]b[\/latex].<\/p>\n<\/div>\n<p>This implies that it does not matter which side of the equation the variable term ends up on.<\/p>\n<p><span style=\"color: #000000;\">The next video includes examples of using the division and multiplication properties to solve equations with the variable on the right side of the equal sign.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex:  Solving One Step Equation by Mult\/Div.  Integers (Var on Right)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/TB1rkPbF8rA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-5.docx\">Transcript-6.1.1-5<\/a><\/h2>\n<h2>Two-Step Linear Equations<\/h2>\n<p><span style=\"color: #000000;\">If the equation is in the form [latex]ax+b=c[\/latex], where [latex]x[\/latex] is the variable, we can solve the equation as before. First we must isolate the [latex]x-[\/latex]term by \u201cundoing\u201d the addition or subtraction. Then we isolate the variable by \u201cundoing\u201d the multiplication or division.<\/span><\/p>\n<div class=\"textbox exercises\">\n<h3><span style=\"color: #000000;\">Example<\/span><\/h3>\n<p><span style=\"color: #000000;\">1. Solve: [latex]4x+6=-14[\/latex]<\/span><\/p>\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\n<p><span style=\"color: #000000;\">In this equation, the variable is only on the left side. It makes sense to call the left side the variable side. Therefore, the right side will be the constant side.<\/span><\/p>\n<table style=\"width: 70%;\" summary=\"The top line says 4x plus 6 equals negative 14.\">\n<tbody>\n<tr>\n<th style=\"width: 1179.02px;\" colspan=\"3\"><span style=\"color: #000000;\">Solving [latex]4x+6=-14[\/latex]<\/span><span style=\"color: #000000;\"><br \/>\n<\/span><\/th>\n<\/tr>\n<tr style=\"height: 45.8594px;\">\n<td style=\"height: 45.8594px;\" colspan=\"2\"><span style=\"color: #000000;\">Since the left side is the variable side, the 6 is out of place. We must &#8220;undo&#8221; adding [latex]6[\/latex] by subtracting [latex]6[\/latex], and to keep the equality we must subtract [latex]6[\/latex] from both sides. Use the Subtraction Property of Equality.<\/span><\/td>\n<td style=\"height: 45.8594px; width: 176px;\"><span style=\"color: #000000;\">[latex]4x+6\\color{red}{-6}=-14\\color{red}{-6}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Simplify.<\/span><\/td>\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]4x=-20[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Now all the [latex]x[\/latex] s are on the left and the constant on the right.<\/span><\/td>\n<td style=\"height: 15px; width: 176px;\"><\/td>\n<\/tr>\n<tr style=\"height: 60px;\">\n<td style=\"height: 60px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Use the Division Property of Equality.<\/span><\/td>\n<td style=\"height: 60px; width: 176px;\"><span style=\"color: #000000;\">[latex]\\Large\\frac{4x}{\\color{red}{4}}\\normalsize =\\Large\\frac{-20}{\\color{red}{4}}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Simplify.<\/span><\/td>\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]x=-5[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Check:<\/span><\/td>\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]4x+6=-14[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><span style=\"color: #000000;\">Let [latex]x=-5[\/latex] .<\/span><\/td>\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]4(\\color{red}{-5})+6=-14[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><\/td>\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]-20+6=-14[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 1179.02px;\" colspan=\"2\"><\/td>\n<td style=\"height: 15px; width: 176px;\"><span style=\"color: #000000;\">[latex]-14=-14\\quad\\checkmark[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #000000;\">2. Solve: [latex]2y - 7=15[\/latex]<\/span><\/p>\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\n<p><span style=\"color: #000000;\">Notice that the variable is only on the left side of the equation, so this will be the variable side, and the right side will be the constant side. Since the left side is the variable side, the [latex]7[\/latex] is out of place. It is subtracted from the [latex]2y[\/latex], so to &#8220;undo&#8221; subtraction, add [latex]7[\/latex] to both sides.<\/span><\/p>\n<table id=\"eip-id1168469592645\" class=\"unnumbered unstyled\" summary=\"The first line says 2y minus 7 equals 15. The left side is labeled\">\n<tbody>\n<tr>\n<th colspan=\"3\">Solving <span style=\"color: #000000;\">[latex]2y - 7=15[\/latex]<\/span><\/th>\n<\/tr>\n<tr>\n<td>[latex]2y-7[\/latex] is the side containing a <span style=\"color: #000000;\"><span style=\"color: #ff0000;\">variable<\/span>.<\/span><span style=\"color: #000000;\">[latex]15[\/latex] is the side containing only a <span style=\"color: #ff0000;\">constant<\/span>.<\/span><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Add [latex]7[\/latex] to both sides.<\/td>\n<td>[latex]2y-7\\color{red}{+7}=15\\color{red}{+7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]2y=22[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"3\">The variables are now on one side and the constants on the other.<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Divide both sides by [latex]2[\/latex].<\/td>\n<td>[latex]\\Large\\frac{2y}{\\color{red}{2}}\\normalsize =\\Large\\frac{22}{\\color{red}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]y=11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Check:<\/td>\n<td>[latex]2y-7=15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Let [latex]y=11[\/latex] .<\/td>\n<td>[latex]2\\cdot\\color{red}{11}-7\\stackrel{\\text{?}}{=}15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>[latex]22-7\\stackrel{\\text{?}}{=}15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>[latex]15=15\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p><span style=\"color: #000000;\">Solve [latex]3y+2=11[\/latex]<\/span><\/p>\n<h4><span style=\"color: #000000;\">Solution<\/span><\/h4>\n<p><span style=\"color: #000000;\">Subtract [latex]2[\/latex] from both sides of the equation to get the term with the variable by itself.<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex]\\displaystyle \\begin{array}{r}3y+2\\,\\,\\,=\\,\\,11\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\color{blue}{-2}\\,\\,\\,\\,\\,\\,\\,\\,\\color{blue}{-2}}\\\\3y\\,\\,\\,\\,=\\,\\,\\,\\,\\,9\\end{array}[\/latex]<\/span><\/p>\n<p><span style=\"color: #000000;\">Divide both sides of the equation by [latex]3[\/latex] to get a coefficient of [latex]1[\/latex] for the variable.<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"color: #000000;\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\underline{3y}\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\underline{9}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,9\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\,\\,\\,\\,=\\,\\,\\,\\,3\\end{array}[\/latex]<\/span><\/p>\n<h4>Answer<\/h4>\n<p><span style=\"color: #000000;\">[latex]y=3[\/latex]<\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try\u00a0It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm142131\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=142131&#38;theme=oea&#38;iframe_resize_id=ohm142131&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #000000;\">In the following video, we show examples of solving two step linear equations.<\/span><\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Solving Two Step Equations (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fCyxSVQKeRw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-content\/uploads\/sites\/5676\/2024\/02\/Transcript-6.1.1-6.docx\">Transcript-6.1.1-6<\/a><\/p>\n<p><span style=\"color: #000000;\">Remember to check the solution of an algebraic equation by substituting the value of the variable into the original equation.<\/span><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":370291,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2846","chapter","type-chapter","status-publish","hentry"],"part":2842,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2846","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2846\/revisions"}],"predecessor-version":[{"id":3068,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2846\/revisions\/3068"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/2842"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2846\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=2846"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2846"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=2846"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=2846"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}