{"id":7551,"date":"2017-04-28T20:00:31","date_gmt":"2017-04-28T20:00:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=7551"},"modified":"2020-10-22T09:06:57","modified_gmt":"2020-10-22T09:06:57","slug":"solve-equations-using-the-subtraction-and-addition-properties-of-equality","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/solve-equations-using-the-subtraction-and-addition-properties-of-equality\/","title":{"raw":"7.1.a - Using the Subtraction and Addition Properties to Solve Equations","rendered":"7.1.a &#8211; Using the Subtraction and Addition Properties to Solve Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine whether a number is a solution to an equation<\/li>\r\n \t<li>Check your solution to a linear equation to verify its accuracy<\/li>\r\n \t<li>Solve equations using the Subtraction and Addition Properties of Equality<\/li>\r\n \t<li>Solve equations that need to be simplified<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe began our work solving equations in previous chapters, where we said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.\u00a0 When you solve an equation, you find the value of the variable that makes the equation true.\r\n\r\nThe simplest type of algebraic equation is a linear equation that has just one variable.\u00a0 We will be solving this type of equation in this section.\u00a0 Specifically, you'll learn how to use the Subtraction and Addition Properties of Equality.\u00a0 \u00a0 But first, we will review how to determine whether a number is a solution of an equation.\r\n<h2>Determine Whether a Number is a Solution of an Equation<\/h2>\r\n<div class=\"textbox shaded\">\r\n<h3 class=\"title\">Solution of an Equation<\/h3>\r\nA solution of an equation is a value of a variable that makes a true statement when substituted into the equation.\r\n\r\nIn the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.\r\n\r\nDetermine whether a number is a solution to an equation.\r\n<ol id=\"eip-95\" class=\"stepwise\">\r\n \t<li>Substitute the number for the variable in the equation.<\/li>\r\n \t<li>Simplify the expressions on both sides of the equation.<\/li>\r\n \t<li>Determine whether the resulting equation is true.\r\n<ul id=\"fs-id1703984\">\r\n \t<li>If it is true, the number is a solution.<\/li>\r\n \t<li>If it is not true, the number is not a solution.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the following example, we will show how to determine whether a number is a solution to an equation that contains addition and subtraction. You can use this idea to check your work later when you are solving equations.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nDetermine whether [latex]y=\\Large\\frac{3}{4}[\/latex] is a solution for [latex]4y+3=8y[\/latex].\r\n\r\nSolution:\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<td><\/td>\r\n<td>[latex]4y+3=8y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{\\Large\\frac{3}{4}}[\/latex] for [latex]y[\/latex]<\/td>\r\n<td>[latex]4(\\color{red}{\\Large\\frac{3}{4}}\\normalsize)+3\\stackrel{\\text{?}}{=}8(\\color{red}{\\Large\\frac{3}{4}})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]3+3\\stackrel{\\text{?}}{=}6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]6=6\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]y=\\Large\\frac{3}{4}[\/latex] results in a true equation, [latex]\\Large\\frac{3}{4}[\/latex] is a solution to the equation [latex]4y+3=8y[\/latex].\r\n\r\n<\/div>\r\nNow it is your turn to determine whether a fraction is the solution to an equation.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141717&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141719&amp;theme=oea&amp;iframe_resize_id=mom3[\/embed]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Subtraction and Addition Properties of Equality<\/h2>\r\n<a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/solving-equations-using-the-subtraction-property-of-equality\/\">We introduced the Subtraction and Addition Properties of Equality earlier by modeling equations with envelopes and counters.<\/a> When you add or subtract the same quantity from both sides of an equation, you still have equality.\u00a0 The image below\u00a0models this with the equation [latex]x+3=8[\/latex].\r\n<figure id=\"CNX_BMath_Figure_08_01_026\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222533\/CNX_BMath_Figure_08_01_026.png\" alt=\"An envelope and three yellow counters are shown on the left side. On the right side are eight yellow counters.\" \/><\/figure>\r\nThe goal is to isolate the variable on one side of the equation. So we \"took away\" [latex]3[\/latex] from both sides of the equation and found the solution [latex]x=5[\/latex].\r\n\r\nSome people picture a balance scale, as in the image below, when they solve equations.\r\n<figure id=\"CNX_BMath_Figure_08_01_001\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222535\/CNX_BMath_Figure_08_01_001.png\" alt=\"Three balance scales are shown. The top scale has one red weight on each side and is balanced. Beside it is \" \/><\/figure>\r\nThe quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other, to keep it balanced.\u00a0 The Addition and Subtraction Properties of Equality\u00a0state that you can add or subtract the same quantity to both sides of an equation, and still maintain an equivalent equation. Sometimes people refer to this as keeping the equation \u201cbalanced.\u201d If you think of an equation as being like a balance scale, the quantities on each side of the equation are equal, or balanced.\r\n\r\nLet\u2019s look at a simple numeric equation, [latex]3+7=10[\/latex], to explore the idea of an equation as being balanced.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image001.jpg#fixme\" alt=\"A balanced scale, with a 3 and a 7 one side and a 10 on the other.\" width=\"318\" height=\"217\" \/>\r\n\r\nThe expressions on each side of the equal sign are equal, so you can add the same value to each side and maintain the equality. Let\u2019s see what happens when [latex]5[\/latex] is added to each side.\r\n<p style=\"text-align: center\">[latex]3+7+5=10+5[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]15=15[\/latex]<\/p>\r\nSince each expression is equal to [latex]15[\/latex], you can see that adding [latex]5[\/latex] to each side of the original equation resulted in a true equation. The equation is still \u201cbalanced.\u201d\r\n\r\nOn the other hand, let\u2019s look at what would happen if you added [latex]5[\/latex] to only one side of the equation.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3+7=10\\\\3+7+5=10\\\\15\\neq 10\\end{array}[\/latex]<\/p>\r\nAdding [latex]5[\/latex] to only one side of the equation resulted in an equation that is false. The equation is no longer \u201cbalanced,\u201d and it is no longer a true equation!\r\n<div class=\"textbox shaded\">\r\n<h3 id=\"fs-id1956928\"><strong>Subtraction Property of Equality<\/strong><\/h3>\r\nFor all real numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex].\r\n\r\nIf two expressions are equal to each other, and you subtract the same value to both sides of the equation, the equation will remain equal.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n<h3 id=\"fs-id1166490863495\"><strong>Addition Property of Equality<\/strong><\/h3>\r\nFor all real numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].\r\n\r\nIf two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.\r\n\r\n<\/div>\r\nIn order to solve an equation, you <b>isolate the variable<\/b>. Isolating the variable means rewriting an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.\u00a0 When you follow the steps to solve an equation, you try to isolate the variable. The variable is a quantity we don't know yet.\u00a0You have a solution when you get the equation [latex]x[\/latex] = some value.\r\n\r\nWhen the equation involves addition or subtraction, use the inverse operation to \u201cundo\u201d the operation in order to isolate the variable. For addition and subtraction, your goal is to change any value being added or subtracted to [latex]0[\/latex], the additive identity.\r\n\r\nIn the following examples, we use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. You can check your solutions by substituting the value into the equation, to make sure you have a true statement.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nSolve: [latex]x+11=-3[\/latex]\r\n[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"190834\"]\r\n\r\nSolution:\r\nTo isolate [latex]x[\/latex], we undo the addition of [latex]11[\/latex] by using the Subtraction Property of Equality.\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 style=\"height: 11px\">\r\n<td style=\"height: 11px;width: 349.656px\" 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\nSince [latex]x=-14[\/latex] makes [latex]x+11=-3[\/latex] a true statement, we know that it is a solution to the equation.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNow you can try solving an equation that requires using the Subtraction Property of Equality.\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\nIn the original equation in the previous example, [latex]11[\/latex] was added to the [latex]x[\/latex] , so we subtracted [latex]11[\/latex] to \"undo\" the addition. In the next example, we will need to \"undo\" subtraction by using the Addition Property of Equality.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nSolve: [latex]m - 4=-5[\/latex]\r\n[reveal-answer q=\"604060\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604060\"]\r\n\r\nSolution:\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<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[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try using the addition property to solve an equation.\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\nIn the following video, we present more examples of solving equations using the addition and subtraction properties.\r\n\r\nhttps:\/\/youtu.be\/yqdlj0lv7Cc\r\n\r\nYou may 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.\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nSolve: [latex]n-\\Large\\frac{3}{8}\\normalsize =\\Large\\frac{1}{2}[\/latex]\r\n[reveal-answer q=\"248157\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"248157\"]\r\n\r\nSolution:\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<td style=\"width: 313.391px\" colspan=\"2\"><\/td>\r\n<td style=\"width: 255.609px\">[latex]n-\\Large\\frac{3}{8}\\normalsize =\\Large\\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-\\Large\\frac{3}{8}\\normalsize\\color{red}{+\\Large\\frac{3}{8}}\\normalsize =\\Large\\frac{1}{2}\\normalsize\\color{red}{+\\Large\\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-\\Large\\frac{3}{8}\\normalsize +\\Large\\frac{3}{8}\\normalsize =\\Large\\frac{4}{8}\\normalsize +\\Large\\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=\\Large\\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-\\Large\\frac{3}{8}\\normalsize =\\Large\\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}{\\Large\\frac{7}{8}}[\/latex]<\/td>\r\n<td style=\"width: 171.391px\">[latex]\\color{red}{\\Large\\frac{7}{8}}\\normalsize -\\Large\\frac{3}{8}\\normalsize\\stackrel{\\text{?}}{=}\\Large\\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]\\Large\\frac{4}{8}\\normalsize\\stackrel{\\text{?}}{=}\\Large\\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]\\Large\\frac{1}{2}\\normalsize =\\Large\\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<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try solving an equation with fractions by using the addition property of equality.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\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\nWatch this video for more examples of solving equations that include fractions and require addition or subtraction.\r\n\r\nhttps:\/\/youtu.be\/KmOvCakGEgM\r\n\r\nYou may encounter equations with decimals, for example in financial or science applications. In the next examples, we will demonstrate how to use the subtraction property of equality to solve equations with decimals.\r\n<div class=\"textbox exercises\">\r\n<h3>eXAMPLE<\/h3>\r\nSolve [latex]a - 3.7=4.3[\/latex]\r\n[reveal-answer q=\"652184\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"652184\"]\r\n\r\nSolution:\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<td style=\"width: 516.391px\" 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[\/hidden-answer]\r\n\r\n<\/div>\r\nNow it is your turn to try solving an equation with decimals by using the addition property of equality.\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\nIn this video, we show more examples of how to solve equations with decimals that require addition and subtraction.\r\n\r\nhttps:\/\/youtu.be\/a6YYJN_bHKs\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Think About It<\/h3>\r\nCan you determine\u00a0what you would do differently if you were asked to solve equations like these?\r\n\r\na) Solve [latex]{12.5}+{ t }= {-7.5}[\/latex].\r\n\r\nWhat makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would solve this equation with decimals.\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n\r\n[reveal-answer q=\"680980\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"680980\"]\r\n\r\nTo solve this equation, you need to remember how to add or subtract decimal numbers. You also need to remember that when you subtract a number from a negative number, your result will be negative.\r\n\r\nUsing the Addition Property of Equality, subtract \u00a0[latex]12.5[\/latex] from both sides of the equation to isolate the variable, \u00a0[latex]t[\/latex]. You choose to\u00a0subtract \u00a0[latex]12.5[\/latex] because \u00a0[latex]12.5[\/latex] is being added to the variable, [latex]t[\/latex].\r\n<p style=\"text-align: center\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\,{12.5}+{t}\\,\\,\\,=\\,\\,\\,\\,{-7.5}\\\\\\,\\,\\,\\,\\underline{-12.5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-12.5}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,t\\,\\,=\\,\\,-20\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">To add two numbers of the same sign,\u00a0first add their absolute values:<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\left|-12.5\\right| = 12.5\\\\\\left|-7.5\\right| = 7.5\\,\\,\\,\\\\12.5 + 7.5 = 20\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">Now apply the sign they share, which is negative:<\/p>\r\n<p style=\"text-align: center\">[latex]-12.5 -7.5 = -20[\/latex]<\/p>\r\n<p style=\"text-align: center\">[\/hidden-answer]<\/p>\r\nb) Solve [latex]\\frac{1}{4} + y = 3[\/latex]. What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would solve this equation with a fraction.\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n\r\n[reveal-answer q=\"690980\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"690980\"]\r\n\r\nUsing the Addition Property of Equality, subtract [latex]\\frac{1}{4}[\/latex] from both sides of the equation to isolate the variable,\u00a0[latex]y[\/latex]. You choose to\u00a0subtract [latex]\\frac{1}{4}[\/latex], as [latex]\\frac{1}{4}[\/latex] is being added to the variable,[latex]y[\/latex].\r\n<p style=\"text-align: center\">[latex]\\displaystyle\\begin{array}{r}\\frac{1}{4} + y\\,\\,\\,=\\,\\,\\,\\,{3}\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\underline{-\\frac{1}{4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-\\frac{1}{4}}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\,\\,=\\,3-\\frac{1}{4}\\,\\,\\end{array}[\/latex]<\/p>\r\nTo subtract\u00a0[latex]\\frac{1}{4}[\/latex] from \u00a0[latex]3[\/latex], you need a common denominator.\r\n\r\nMake \u00a0[latex]3[\/latex] into a fraction by dividing by \u00a0[latex]1[\/latex],\u00a0[latex]\\frac{3}{1}[\/latex]. \u00a0Your denominators are \u00a0[latex]1[\/latex] and \u00a0[latex]4[\/latex]. The least common multiple is \u00a0[latex]4[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\frac{3}{1}\\cdot\\frac{4}{4}=\\frac{12}{4}\\\\\\frac{12}{4} -\\frac{1}{4} =\\frac{11}{4}\\end{array}[\/latex]<\/p>\r\nTherefore, [latex]y = 3 -\\frac{1}{4} =\\frac{11}{4}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Solving Equations that Need to be Simplified<\/h2>\r\nThe examples above are sometimes called <b>one-step equations<\/b> because they require only one step to solve. In these examples, you either added or subtracted a <b>constant<\/b> from both sides of the equation to isolate the variable and solve the equation.\r\nIn the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve:\r\n\r\n[latex]3x - 7 - 2x - 4=1[\/latex]\r\n\r\nSolution:\r\nThe left side of the equation has an expression that we should simplify before trying to isolate the variable.\r\n<table id=\"eip-id1168467171254\" class=\"unnumbered unstyled\" summary=\"The first line shows the equation 3x minus 7 minus 2x minus 4 equals 1. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3x-7-2x-4=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange the terms, using the Commutative Property of Addition.<\/td>\r\n<td>[latex]3x-2x-7-4=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]x-11=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]11[\/latex] to both sides to isolate [latex]x[\/latex] .<\/td>\r\n<td>[latex]x-11\\color{red}{+11}=1\\color{red}{+11}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.Substitute [latex]x=12[\/latex] into the original equation.\r\n[latex]3x-7-2x-4=1[\/latex][latex]3(\\color{red}{12})-7-2(\\color{red}{12})-4=1[\/latex][latex]36-7-24-4=1[\/latex][latex]29-24-4=1[\/latex][latex]5-4=1[\/latex][latex]1=1\\quad\\checkmark[\/latex]The solution checks.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNow you can try solving a couple\u00a0of equations where you should simplify first.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141735&amp;theme=oea&amp;iframe_resize_id=mom2[\/embed]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine whether a number is a solution to an equation<\/li>\n<li>Check your solution to a linear equation to verify its accuracy<\/li>\n<li>Solve equations using the Subtraction and Addition Properties of Equality<\/li>\n<li>Solve equations that need to be simplified<\/li>\n<\/ul>\n<\/div>\n<p>We began our work solving equations in previous chapters, where we said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.\u00a0 When you solve an equation, you find the value of the variable that makes the equation true.<\/p>\n<p>The simplest type of algebraic equation is a linear equation that has just one variable.\u00a0 We will be solving this type of equation in this section.\u00a0 Specifically, you&#8217;ll learn how to use the Subtraction and Addition Properties of Equality.\u00a0 \u00a0 But first, we will review how to determine whether a number is a solution of an equation.<\/p>\n<h2>Determine Whether a Number is a Solution of an Equation<\/h2>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Solution of an Equation<\/h3>\n<p>A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.<\/p>\n<p>In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.<\/p>\n<p>Determine whether a number is a solution to an equation.<\/p>\n<ol id=\"eip-95\" class=\"stepwise\">\n<li>Substitute the number for the variable in the equation.<\/li>\n<li>Simplify the expressions on both sides of the equation.<\/li>\n<li>Determine whether the resulting equation is true.\n<ul id=\"fs-id1703984\">\n<li>If it is true, the number is a solution.<\/li>\n<li>If it is not true, the number is not a solution.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>In the following example, we will show how to determine whether a number is a solution to an equation that contains addition and subtraction. You can use this idea to check your work later when you are solving equations.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Determine whether [latex]y=\\Large\\frac{3}{4}[\/latex] is a solution for [latex]4y+3=8y[\/latex].<\/p>\n<p>Solution:<\/p>\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<td><\/td>\n<td>[latex]4y+3=8y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{\\Large\\frac{3}{4}}[\/latex] for [latex]y[\/latex]<\/td>\n<td>[latex]4(\\color{red}{\\Large\\frac{3}{4}}\\normalsize)+3\\stackrel{\\text{?}}{=}8(\\color{red}{\\Large\\frac{3}{4}})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]3+3\\stackrel{\\text{?}}{=}6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]6=6\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]y=\\Large\\frac{3}{4}[\/latex] results in a true equation, [latex]\\Large\\frac{3}{4}[\/latex] is a solution to the equation [latex]4y+3=8y[\/latex].<\/p>\n<\/div>\n<p>Now it is your turn to determine whether a fraction is the solution to an equation.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141717\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141717&#38;theme=oea&#38;iframe_resize_id=ohm141717&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm141719\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141719&#38;theme=oea&#38;iframe_resize_id=ohm141719&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Subtraction and Addition Properties of Equality<\/h2>\n<p><a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/solving-equations-using-the-subtraction-property-of-equality\/\">We introduced the Subtraction and Addition Properties of Equality earlier by modeling equations with envelopes and counters.<\/a> When you add or subtract the same quantity from both sides of an equation, you still have equality.\u00a0 The image below\u00a0models this with the equation [latex]x+3=8[\/latex].<\/p>\n<figure id=\"CNX_BMath_Figure_08_01_026\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222533\/CNX_BMath_Figure_08_01_026.png\" alt=\"An envelope and three yellow counters are shown on the left side. On the right side are eight yellow counters.\" \/><\/figure>\n<p>The goal is to isolate the variable on one side of the equation. So we &#8220;took away&#8221; [latex]3[\/latex] from both sides of the equation and found the solution [latex]x=5[\/latex].<\/p>\n<p>Some people picture a balance scale, as in the image below, when they solve equations.<\/p>\n<figure id=\"CNX_BMath_Figure_08_01_001\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24222535\/CNX_BMath_Figure_08_01_001.png\" alt=\"Three balance scales are shown. The top scale has one red weight on each side and is balanced. Beside it is\" \/><\/figure>\n<p>The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other, to keep it balanced.\u00a0 The Addition and Subtraction Properties of Equality\u00a0state that you can add or subtract the same quantity to both sides of an equation, and still maintain an equivalent equation. Sometimes people refer to this as keeping the equation \u201cbalanced.\u201d If you think of an equation as being like a balance scale, the quantities on each side of the equation are equal, or balanced.<\/p>\n<p>Let\u2019s look at a simple numeric equation, [latex]3+7=10[\/latex], to explore the idea of an equation as being balanced.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image001.jpg#fixme\" alt=\"A balanced scale, with a 3 and a 7 one side and a 10 on the other.\" width=\"318\" height=\"217\" \/><\/p>\n<p>The expressions on each side of the equal sign are equal, so you can add the same value to each side and maintain the equality. Let\u2019s see what happens when [latex]5[\/latex] is added to each side.<\/p>\n<p style=\"text-align: center\">[latex]3+7+5=10+5[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]15=15[\/latex]<\/p>\n<p>Since each expression is equal to [latex]15[\/latex], you can see that adding [latex]5[\/latex] to each side of the original equation resulted in a true equation. The equation is still \u201cbalanced.\u201d<\/p>\n<p>On the other hand, let\u2019s look at what would happen if you added [latex]5[\/latex] to only one side of the equation.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3+7=10\\\\3+7+5=10\\\\15\\neq 10\\end{array}[\/latex]<\/p>\n<p>Adding [latex]5[\/latex] to only one side of the equation resulted in an equation that is false. The equation is no longer \u201cbalanced,\u201d and it is no longer a true equation!<\/p>\n<div class=\"textbox shaded\">\n<h3 id=\"fs-id1956928\"><strong>Subtraction Property of Equality<\/strong><\/h3>\n<p>For all real numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex].<\/p>\n<p>If two expressions are equal to each other, and you subtract the same value to both sides of the equation, the equation will remain equal.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<h3 id=\"fs-id1166490863495\"><strong>Addition Property of Equality<\/strong><\/h3>\n<p>For all real numbers [latex]a,b[\/latex], and [latex]c[\/latex], if [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].<\/p>\n<p>If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.<\/p>\n<\/div>\n<p>In order to solve an equation, you <b>isolate the variable<\/b>. Isolating the variable means rewriting an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.\u00a0 When you follow the steps to solve an equation, you try to isolate the variable. The variable is a quantity we don&#8217;t know yet.\u00a0You have a solution when you get the equation [latex]x[\/latex] = some value.<\/p>\n<p>When the equation involves addition or subtraction, use the inverse operation to \u201cundo\u201d the operation in order to isolate the variable. For addition and subtraction, your goal is to change any value being added or subtracted to [latex]0[\/latex], the additive identity.<\/p>\n<p>In the following examples, we use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. You can check your solutions by substituting the value into the equation, to make sure you have a true statement.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Solve: [latex]x+11=-3[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q190834\">Show Solution<\/span><\/p>\n<div id=\"q190834\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nTo isolate [latex]x[\/latex], we undo the addition of [latex]11[\/latex] by using the Subtraction Property of Equality.<\/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 style=\"height: 11px\">\n<td style=\"height: 11px;width: 349.656px\" 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>Since [latex]x=-14[\/latex] makes [latex]x+11=-3[\/latex] a true statement, we know that it is a solution to the equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Now you can try solving an equation that requires using the Subtraction Property of Equality.<\/p>\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<p>In the original equation in the previous example, [latex]11[\/latex] was added to the [latex]x[\/latex] , so we subtracted [latex]11[\/latex] to &#8220;undo&#8221; the addition. In the next example, we will need to &#8220;undo&#8221; subtraction by using the Addition Property of Equality.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Solve: [latex]m - 4=-5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q604060\">Show Solution<\/span><\/p>\n<div id=\"q604060\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\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<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>\n<\/div>\n<p>Now you can try using the addition property to solve an equation.<\/p>\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>In the following video, we present more examples of solving equations using the addition and subtraction properties.<\/p>\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<p>You may 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.<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Solve: [latex]n-\\Large\\frac{3}{8}\\normalsize =\\Large\\frac{1}{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q248157\">Show Solution<\/span><\/p>\n<div id=\"q248157\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\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<td style=\"width: 313.391px\" colspan=\"2\"><\/td>\n<td style=\"width: 255.609px\">[latex]n-\\Large\\frac{3}{8}\\normalsize =\\Large\\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-\\Large\\frac{3}{8}\\normalsize\\color{red}{+\\Large\\frac{3}{8}}\\normalsize =\\Large\\frac{1}{2}\\normalsize\\color{red}{+\\Large\\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-\\Large\\frac{3}{8}\\normalsize +\\Large\\frac{3}{8}\\normalsize =\\Large\\frac{4}{8}\\normalsize +\\Large\\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=\\Large\\frac{7}{8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 142px\">Check:<\/td>\n<td style=\"width: 171.391px\">[latex]n-\\Large\\frac{3}{8}\\normalsize =\\Large\\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}{\\Large\\frac{7}{8}}[\/latex]<\/td>\n<td style=\"width: 171.391px\">[latex]\\color{red}{\\Large\\frac{7}{8}}\\normalsize -\\Large\\frac{3}{8}\\normalsize\\stackrel{\\text{?}}{=}\\Large\\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]\\Large\\frac{4}{8}\\normalsize\\stackrel{\\text{?}}{=}\\Large\\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]\\Large\\frac{1}{2}\\normalsize =\\Large\\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<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try solving an equation with fractions by using the addition property of equality.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\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>Watch this video for more examples of solving equations that include fractions and require addition or subtraction.<\/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<p>You may encounter equations with decimals, for example in financial or science applications. In the next examples, we will demonstrate how to use the subtraction property of equality to solve equations with decimals.<\/p>\n<div class=\"textbox exercises\">\n<h3>eXAMPLE<\/h3>\n<p>Solve [latex]a - 3.7=4.3[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q652184\">Show Solution<\/span><\/p>\n<div id=\"q652184\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\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<td style=\"width: 516.391px\" 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>\n<\/div>\n<p>Now it is your turn to try solving an equation with decimals by using the addition property of equality.<\/p>\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>In this video, we show more examples of how to solve equations with decimals that require addition and subtraction.<\/p>\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<div class=\"bcc-box bcc-success\">\n<h3>Think About It<\/h3>\n<p>Can you determine\u00a0what you would do differently if you were asked to solve equations like these?<\/p>\n<p>a) Solve [latex]{12.5}+{ t }= {-7.5}[\/latex].<\/p>\n<p>What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would solve this equation with decimals.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q680980\">Show Solution<\/span><\/p>\n<div id=\"q680980\" class=\"hidden-answer\" style=\"display: none\">\n<p>To solve this equation, you need to remember how to add or subtract decimal numbers. You also need to remember that when you subtract a number from a negative number, your result will be negative.<\/p>\n<p>Using the Addition Property of Equality, subtract \u00a0[latex]12.5[\/latex] from both sides of the equation to isolate the variable, \u00a0[latex]t[\/latex]. You choose to\u00a0subtract \u00a0[latex]12.5[\/latex] because \u00a0[latex]12.5[\/latex] is being added to the variable, [latex]t[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\,{12.5}+{t}\\,\\,\\,=\\,\\,\\,\\,{-7.5}\\\\\\,\\,\\,\\,\\underline{-12.5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-12.5}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,t\\,\\,=\\,\\,-20\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">To add two numbers of the same sign,\u00a0first add their absolute values:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\left|-12.5\\right| = 12.5\\\\\\left|-7.5\\right| = 7.5\\,\\,\\,\\\\12.5 + 7.5 = 20\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">Now apply the sign they share, which is negative:<\/p>\n<p style=\"text-align: center\">[latex]-12.5 -7.5 = -20[\/latex]<\/p>\n<p style=\"text-align: center\"><\/div>\n<\/div>\n<p>b) Solve [latex]\\frac{1}{4} + y = 3[\/latex]. What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would solve this equation with a fraction.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q690980\">Show Solution<\/span><\/p>\n<div id=\"q690980\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using the Addition Property of Equality, subtract [latex]\\frac{1}{4}[\/latex] from both sides of the equation to isolate the variable,\u00a0[latex]y[\/latex]. You choose to\u00a0subtract [latex]\\frac{1}{4}[\/latex], as [latex]\\frac{1}{4}[\/latex] is being added to the variable,[latex]y[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle\\begin{array}{r}\\frac{1}{4} + y\\,\\,\\,=\\,\\,\\,\\,{3}\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\underline{-\\frac{1}{4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-\\frac{1}{4}}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\,\\,=\\,3-\\frac{1}{4}\\,\\,\\end{array}[\/latex]<\/p>\n<p>To subtract\u00a0[latex]\\frac{1}{4}[\/latex] from \u00a0[latex]3[\/latex], you need a common denominator.<\/p>\n<p>Make \u00a0[latex]3[\/latex] into a fraction by dividing by \u00a0[latex]1[\/latex],\u00a0[latex]\\frac{3}{1}[\/latex]. \u00a0Your denominators are \u00a0[latex]1[\/latex] and \u00a0[latex]4[\/latex]. The least common multiple is \u00a0[latex]4[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\frac{3}{1}\\cdot\\frac{4}{4}=\\frac{12}{4}\\\\\\frac{12}{4} -\\frac{1}{4} =\\frac{11}{4}\\end{array}[\/latex]<\/p>\n<p>Therefore, [latex]y = 3 -\\frac{1}{4} =\\frac{11}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Solving Equations that Need to be Simplified<\/h2>\n<p>The examples above are sometimes called <b>one-step equations<\/b> because they require only one step to solve. In these examples, you either added or subtracted a <b>constant<\/b> from both sides of the equation to isolate the variable and solve the equation.<br \/>\nIn the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve:<\/p>\n<p>[latex]3x - 7 - 2x - 4=1[\/latex]<\/p>\n<p>Solution:<br \/>\nThe left side of the equation has an expression that we should simplify before trying to isolate the variable.<\/p>\n<table id=\"eip-id1168467171254\" class=\"unnumbered unstyled\" summary=\"The first line shows the equation 3x minus 7 minus 2x minus 4 equals 1. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3x-7-2x-4=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rearrange the terms, using the Commutative Property of Addition.<\/td>\n<td>[latex]3x-2x-7-4=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]x-11=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add [latex]11[\/latex] to both sides to isolate [latex]x[\/latex] .<\/td>\n<td>[latex]x-11\\color{red}{+11}=1\\color{red}{+11}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.Substitute [latex]x=12[\/latex] into the original equation.<br \/>\n[latex]3x-7-2x-4=1[\/latex][latex]3(\\color{red}{12})-7-2(\\color{red}{12})-4=1[\/latex][latex]36-7-24-4=1[\/latex][latex]29-24-4=1[\/latex][latex]5-4=1[\/latex][latex]1=1\\quad\\checkmark[\/latex]The solution checks.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Now you can try solving a couple\u00a0of equations where you should simplify first.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141735\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141735&#38;theme=oea&#38;iframe_resize_id=ohm141735&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-7551\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 141717, 141719, 141721, 141736, 141732. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License, CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Solve One Step Equations By Add and Subtract Whole Numbers (Variable on Left). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/yqdlj0lv7Cc\">https:\/\/youtu.be\/yqdlj0lv7Cc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Solve One Step Equations With Fraction by Adding or Subtracting. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KmOvCakGEgM\">https:\/\/youtu.be\/KmOvCakGEgM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Solve a One Step Equation With Decimals by Adding and Subtracting. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/a6YYJN_bHKs\">https:\/\/youtu.be\/a6YYJN_bHKs<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Solve One Step Equations By Add and Subtract Whole Numbers (Variable on Left)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/yqdlj0lv7Cc\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Solve One Step Equations With Fraction by Adding or Subtracting\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/KmOvCakGEgM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Solve a One Step Equation With Decimals by Adding and Subtracting\",\"author\":\"James Sousa 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