{"id":3853,"date":"2020-02-09T20:19:56","date_gmt":"2020-02-09T20:19:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3853"},"modified":"2024-06-26T18:03:37","modified_gmt":"2024-06-26T18:03:37","slug":"using-the-properties-of-equality-to-solve-equations-with-decimals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/using-the-properties-of-equality-to-solve-equations-with-decimals\/","title":{"raw":"Solve Equations With Decimals","rendered":"Solve Equations With Decimals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine whether a decimal is a solution to an equation<\/li>\r\n \t<li>Solve an equation that contains decimals using the addition and subtraction properties<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Determine Whether a Decimal is a Solution of an Equation<\/h2>\r\nSolving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family\u2019s budget, or planning for the future of your business, you\u2019ll be solving equations with decimals.\r\n\r\nNow that we\u2019ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal. We\u2019ll list these steps here again for easy reference.\r\n<div class=\"textbox shaded\">\r\n<h3>Determine whether a number is a solution to an equation<\/h3>\r\n<ol id=\"eip-id1168469438337\" 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-id2482544\">\r\n \t<li>If so, the number is a solution.<\/li>\r\n \t<li>If not, the number is not a solution.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDetermine whether each of the following is a solution of [latex]x - 0.7=1.5[\/latex]\r\n\r\n1.\u00a0 [latex]x=1[\/latex]\r\n\r\n2.\u00a0 [latex]x=-0.8[\/latex]\r\n\r\n3.\u00a0 [latex]x=2.2[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168466079883\" class=\"unnumbered unstyled\" summary=\"The top line says x minus 0.7 equals 1.5. The next line says to substitute a red 1 for x. The image shows a red 1 minus 0.7 followed by an equal sign with a question mark and then 1.5. The last line says to subtract and shows that 0.3 does not equal 1.5.\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{1}[\/latex] for x.<\/td>\r\n<td>[latex]\\color{red}{1} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]0.3\\not=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]x=1[\/latex] does not result in a true equation, [latex]1[\/latex] is not a solution to the equation.\r\n<table id=\"eip-id1168468762265\" class=\"unnumbered unstyled\" summary=\"The top line says x minus 0.7 equals 1.5. The next line says to substitute a red negative 0.8 for x. The image shows a red negative 0.8 minus 0.7 followed by an equal sign with a question mark and then 1.5. The last line says to subtract and shows that negative 1.5 does not equal 1.5.\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{0.8}[\/latex] for x.<\/td>\r\n<td>[latex]\\color{red}{0.8} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]-1.5\\not=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]x=-0.8[\/latex] does not result in a true equation, [latex]-0.8[\/latex] is not a solution to the equation.\r\n<table id=\"eip-id1168466232209\" class=\"unnumbered unstyled\" summary=\"The top line says x minus 0.7 equals 1.5. The next line says to substitute a red 2.2 for x. The image shows a red 2.2 minus 0.7 followed by an equal sign with a question mark and then 1.5. The last line says to subtract and shows that 1.5 equals 1.5, followed by a check mark.\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{2.2}[\/latex] for x.<\/td>\r\n<td>[latex]\\color{red}{2.2} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]1.5=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]x=2.2[\/latex] results in a true equation, [latex]2.2[\/latex] is a solution to the equation.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146375[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Solve Equations with Decimals<\/h2>\r\nIn previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.\r\n<div class=\"textbox shaded\">\r\n<h3>Properties of Equality<\/h3>\r\n<table id=\"fs-id1752382\" class=\"unnumbered\" summary=\"A table is shown with 2 rows and 2 columns. The first cell says, \">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>Subtraction Property of Equality<\/strong>\r\n\r\nFor any numbers [latex]a,b,\\text{and }c[\/latex]\r\n\r\nIf [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex].<\/td>\r\n<td><strong>Addition Property of Equality<\/strong>\r\n\r\nFor any numbers [latex]a,b,\\text{and }c[\/latex]\r\n\r\nIf [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>The Division Property of Equality<\/strong>\r\n\r\nFor any numbers [latex]a,b,\\text{and }c,\\text{and }c\\ne 0[\/latex]\r\n\r\nIf [latex]a=b[\/latex], then [latex]{\\Large\\frac{a}{c}}={\\Large\\frac{b}{c}}[\/latex]<\/td>\r\n<td><strong>The Multiplication Property of Equality<\/strong>\r\n\r\nFor any numbers [latex]a,b,\\text{and }c[\/latex]\r\n\r\nIf [latex]a=b[\/latex], then [latex]ac=bc[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nWhen you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]y+2.3=-4.7[\/latex]\r\n[reveal-answer q=\"908331\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"908331\"]\r\n\r\nSolution\r\nWe will use the Subtraction Property of Equality to isolate the variable.\r\n<table id=\"eip-id1168469371657\" class=\"unnumbered unstyled\" summary=\"The top line says y plus 2.3 equals negative 4.7. The next line says to subtract a red 2.3 from each side, to undo the addition. Y plus 2.3 minus a red 2.4 equals negative 4.7 minus a red 2.3. The next line says to simplify and shows y equals negative 7. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>[latex]y+2.3=-4.7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Subtract [latex]\\color{red}{2.3}[\/latex] from each side, to undo the addition.<\/td>\r\n<td>[latex]y+2.3\\color{red}{- 2.3}=-4.7\\color{red}{- 2.3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]y-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Check:<\/strong><\/td>\r\n<td>[latex]y+2.3=-4.7[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]y=\\color{red}{-7}[\/latex].<\/td>\r\n<td>[latex]\\color{red}{-7}+2.3\\stackrel{?}{=}-4.7[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-4.7=-4.7[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]y=-7[\/latex] makes [latex]y+2.3=-4.7[\/latex] a true statement, we know we have found a solution to this equation.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146377[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]a - 4.75=-1.39[\/latex]\r\n[reveal-answer q=\"265677\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"265677\"]\r\n\r\nSolution\r\nWe will use the Addition Property of Equality.\r\n<table id=\"eip-id1168469858838\" class=\"unnumbered unstyled\" summary=\"The top line says a minus 4.75 equals negative 1.39. The next line says to add a red 4.75 to each side, to undo the subtraction. a minus 4.75 plus a red 4.75 equals negative 1.39 plus a red 4.75. The next line says to simplify and shows a equals 3.36. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>[latex]a-4.75=-1.39[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Add [latex]4.75[\/latex] to each side, to undo the subtraction.<\/td>\r\n<td>[latex]a-4.75+\\color{red}{4.75}=-1.39+\\color{red}{4.75}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]a=3.36[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Check:<\/strong><\/td>\r\n<td>[latex]a-4.75=-1.39[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]a=\\color{red}{3.36}[\/latex].<\/td>\r\n<td>[latex]\\color{red}{3.36}-4.75\\stackrel{?}{=}-1.39[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-1.39=-1.39[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince the result is a true statement, [latex]a=3.36[\/latex] is a solution to the equation.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146379[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show more examples of using the addition and subtraction properties of equality to solve linear equations that contain decimals.\r\n\r\nhttps:\/\/youtu.be\/D8wKGlxf6bM\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]-4.8=0.8n[\/latex]\r\n[reveal-answer q=\"249570\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"249570\"]\r\n\r\nSolution\r\nWe will use the Division Property of Equality.\r\nUse the Properties of Equality to find a value for [latex]n[\/latex].\r\n<table id=\"eip-id1168468652514\" class=\"unnumbered unstyled\" summary=\"The first line says negative 4.8 equals 0.8 times n. The next line says, \">\r\n<tbody>\r\n<tr style=\"height: 15.46875px;\">\r\n<td style=\"height: 15.46875px;\" colspan=\"2\"><\/td>\r\n<td style=\"height: 15.46875px;\">[latex]-4.8=0.8n[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 43px;\">\r\n<td style=\"height: 43px;\" colspan=\"2\">We must divide both sides by [latex]0.8[\/latex] to isolate <em>n<\/em>.<\/td>\r\n<td style=\"height: 43px;\">[latex]{\\Large\\frac{-4.8}{\\color{red}{0.8}}}={\\Large\\frac{0.8n}{\\color{red}{0.8}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 25px;\">\r\n<td style=\"height: 25px;\" colspan=\"2\">Simplify.<\/td>\r\n<td style=\"height: 25px;\">[latex]-6=n[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"height: 22px;\"><strong>Check:<\/strong><\/td>\r\n<td style=\"height: 22px;\">[latex]-4.8=0.8n[\/latex]<\/td>\r\n<td style=\"height: 22px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"height: 28px;\">Substitute [latex]n=\\color{red}{-6}[\/latex].<\/td>\r\n<td style=\"height: 28px;\">[latex]-4.8\\stackrel{?}{=}0.8(\\color{red}{-6})[\/latex]<\/td>\r\n<td style=\"height: 28px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px;\"><\/td>\r\n<td style=\"height: 24px;\">[latex]-4.8=-4.8[\/latex]<\/td>\r\n<td style=\"height: 24px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]n=-6[\/latex] makes [latex]-4.8=0.8n[\/latex] a true statement, we know we have a solution.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146382[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the next video to see how to solve another equation with decimals that requires division.\r\n\r\nhttps:\/\/youtu.be\/paV-sXbRkHA\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]\r\n[reveal-answer q=\"571663\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"571663\"]\r\n\r\nSolution\r\nWe will use the Multiplication Property of Equality.\r\n<table id=\"eip-id1168468534925\" class=\"unnumbered unstyled\" summary=\"The first line says p divided by negative 1.8 equals negative 6.5. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\"><\/td>\r\n<td>[latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Here, <em>p<\/em> is divided by [latex]\u22121.8[\/latex]. We must multiply by [latex]\u22121.8[\/latex] to isolate <em>p<\/em><\/td>\r\n<td>[latex]\\color{red}{-1.8}({\\Large\\frac{p}{-1.8}})=\\color{red}{-1.8}(-6.5)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Multiply.<\/td>\r\n<td>[latex]p=11.7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Check:<\/strong><\/td>\r\n<td>[latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\Large\\frac{\\color{red}{11.7}}{-1.8}}\\stackrel{?}{=}-6.5[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0Substitute [latex]p=\\color{red}{11.7}[\/latex].<\/td>\r\n<td>[latex]-6.5=-6.5[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nA solution to [latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex] is [latex]p=11.7[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146383[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video shows an example of how to solve an equation with decimals that requires multiplication.\r\n\r\nhttps:\/\/youtu.be\/q2s9GaEwSi8","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine whether a decimal is a solution to an equation<\/li>\n<li>Solve an equation that contains decimals using the addition and subtraction properties<\/li>\n<\/ul>\n<\/div>\n<h2>Determine Whether a Decimal is a Solution of an Equation<\/h2>\n<p>Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family\u2019s budget, or planning for the future of your business, you\u2019ll be solving equations with decimals.<\/p>\n<p>Now that we\u2019ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal. We\u2019ll list these steps here again for easy reference.<\/p>\n<div class=\"textbox shaded\">\n<h3>Determine whether a number is a solution to an equation<\/h3>\n<ol id=\"eip-id1168469438337\" 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-id2482544\">\n<li>If so, the number is a solution.<\/li>\n<li>If not, the number is not a solution.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Determine whether each of the following is a solution of [latex]x - 0.7=1.5[\/latex]<\/p>\n<p>1.\u00a0 [latex]x=1[\/latex]<\/p>\n<p>2.\u00a0 [latex]x=-0.8[\/latex]<\/p>\n<p>3.\u00a0 [latex]x=2.2[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168466079883\" class=\"unnumbered unstyled\" summary=\"The top line says x minus 0.7 equals 1.5. The next line says to substitute a red 1 for x. The image shows a red 1 minus 0.7 followed by an equal sign with a question mark and then 1.5. The last line says to subtract and shows that 0.3 does not equal 1.5.\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{1}[\/latex] for x.<\/td>\n<td>[latex]\\color{red}{1} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]0.3\\not=1.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]x=1[\/latex] does not result in a true equation, [latex]1[\/latex] is not a solution to the equation.<\/p>\n<table id=\"eip-id1168468762265\" class=\"unnumbered unstyled\" summary=\"The top line says x minus 0.7 equals 1.5. The next line says to substitute a red negative 0.8 for x. The image shows a red negative 0.8 minus 0.7 followed by an equal sign with a question mark and then 1.5. The last line says to subtract and shows that negative 1.5 does not equal 1.5.\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{0.8}[\/latex] for x.<\/td>\n<td>[latex]\\color{red}{0.8} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]-1.5\\not=1.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]x=-0.8[\/latex] does not result in a true equation, [latex]-0.8[\/latex] is not a solution to the equation.<\/p>\n<table id=\"eip-id1168466232209\" class=\"unnumbered unstyled\" summary=\"The top line says x minus 0.7 equals 1.5. The next line says to substitute a red 2.2 for x. The image shows a red 2.2 minus 0.7 followed by an equal sign with a question mark and then 1.5. The last line says to subtract and shows that 1.5 equals 1.5, followed by a check mark.\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{2.2}[\/latex] for x.<\/td>\n<td>[latex]\\color{red}{2.2} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]1.5=1.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]x=2.2[\/latex] results in a true equation, [latex]2.2[\/latex] is a solution to the equation.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146375\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146375&theme=oea&iframe_resize_id=ohm146375&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Solve Equations with Decimals<\/h2>\n<p>In previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.<\/p>\n<div class=\"textbox shaded\">\n<h3>Properties of Equality<\/h3>\n<table id=\"fs-id1752382\" class=\"unnumbered\" summary=\"A table is shown with 2 rows and 2 columns. The first cell says,\">\n<tbody>\n<tr valign=\"top\">\n<td><strong>Subtraction Property of Equality<\/strong><\/p>\n<p>For any numbers [latex]a,b,\\text{and }c[\/latex]<\/p>\n<p>If [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex].<\/td>\n<td><strong>Addition Property of Equality<\/strong><\/p>\n<p>For any numbers [latex]a,b,\\text{and }c[\/latex]<\/p>\n<p>If [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>The Division Property of Equality<\/strong><\/p>\n<p>For any numbers [latex]a,b,\\text{and }c,\\text{and }c\\ne 0[\/latex]<\/p>\n<p>If [latex]a=b[\/latex], then [latex]{\\Large\\frac{a}{c}}={\\Large\\frac{b}{c}}[\/latex]<\/td>\n<td><strong>The Multiplication Property of Equality<\/strong><\/p>\n<p>For any numbers [latex]a,b,\\text{and }c[\/latex]<\/p>\n<p>If [latex]a=b[\/latex], then [latex]ac=bc[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]y+2.3=-4.7[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q908331\">Show Solution<\/span><\/p>\n<div id=\"q908331\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nWe will use the Subtraction Property of Equality to isolate the variable.<\/p>\n<table id=\"eip-id1168469371657\" class=\"unnumbered unstyled\" summary=\"The top line says y plus 2.3 equals negative 4.7. The next line says to subtract a red 2.3 from each side, to undo the addition. Y plus 2.3 minus a red 2.4 equals negative 4.7 minus a red 2.3. The next line says to simplify and shows y equals negative 7. The next line says,\">\n<tbody>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>[latex]y+2.3=-4.7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Subtract [latex]\\color{red}{2.3}[\/latex] from each side, to undo the addition.<\/td>\n<td>[latex]y+2.3\\color{red}{- 2.3}=-4.7\\color{red}{- 2.3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]y-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Check:<\/strong><\/td>\n<td>[latex]y+2.3=-4.7[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]y=\\color{red}{-7}[\/latex].<\/td>\n<td>[latex]\\color{red}{-7}+2.3\\stackrel{?}{=}-4.7[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-4.7=-4.7[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]y=-7[\/latex] makes [latex]y+2.3=-4.7[\/latex] a true statement, we know we have found a solution to this equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146377\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146377&theme=oea&iframe_resize_id=ohm146377&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]a - 4.75=-1.39[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q265677\">Show Solution<\/span><\/p>\n<div id=\"q265677\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nWe will use the Addition Property of Equality.<\/p>\n<table id=\"eip-id1168469858838\" class=\"unnumbered unstyled\" summary=\"The top line says a minus 4.75 equals negative 1.39. The next line says to add a red 4.75 to each side, to undo the subtraction. a minus 4.75 plus a red 4.75 equals negative 1.39 plus a red 4.75. The next line says to simplify and shows a equals 3.36. The next line says,\">\n<tbody>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>[latex]a-4.75=-1.39[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Add [latex]4.75[\/latex] to each side, to undo the subtraction.<\/td>\n<td>[latex]a-4.75+\\color{red}{4.75}=-1.39+\\color{red}{4.75}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]a=3.36[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Check:<\/strong><\/td>\n<td>[latex]a-4.75=-1.39[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]a=\\color{red}{3.36}[\/latex].<\/td>\n<td>[latex]\\color{red}{3.36}-4.75\\stackrel{?}{=}-1.39[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-1.39=-1.39[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since the result is a true statement, [latex]a=3.36[\/latex] is a solution to the equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146379\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146379&theme=oea&iframe_resize_id=ohm146379&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of using the addition and subtraction properties of equality to solve linear equations that contain decimals.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solving One Step Equations Using Addition and Subtraction (Decimals)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/D8wKGlxf6bM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]-4.8=0.8n[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q249570\">Show Solution<\/span><\/p>\n<div id=\"q249570\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nWe will use the Division Property of Equality.<br \/>\nUse the Properties of Equality to find a value for [latex]n[\/latex].<\/p>\n<table id=\"eip-id1168468652514\" class=\"unnumbered unstyled\" summary=\"The first line says negative 4.8 equals 0.8 times n. The next line says,\">\n<tbody>\n<tr style=\"height: 15.46875px;\">\n<td style=\"height: 15.46875px;\" colspan=\"2\"><\/td>\n<td style=\"height: 15.46875px;\">[latex]-4.8=0.8n[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 43px;\">\n<td style=\"height: 43px;\" colspan=\"2\">We must divide both sides by [latex]0.8[\/latex] to isolate <em>n<\/em>.<\/td>\n<td style=\"height: 43px;\">[latex]{\\Large\\frac{-4.8}{\\color{red}{0.8}}}={\\Large\\frac{0.8n}{\\color{red}{0.8}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 25px;\">\n<td style=\"height: 25px;\" colspan=\"2\">Simplify.<\/td>\n<td style=\"height: 25px;\">[latex]-6=n[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"height: 22px;\"><strong>Check:<\/strong><\/td>\n<td style=\"height: 22px;\">[latex]-4.8=0.8n[\/latex]<\/td>\n<td style=\"height: 22px;\"><\/td>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"height: 28px;\">Substitute [latex]n=\\color{red}{-6}[\/latex].<\/td>\n<td style=\"height: 28px;\">[latex]-4.8\\stackrel{?}{=}0.8(\\color{red}{-6})[\/latex]<\/td>\n<td style=\"height: 28px;\"><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px;\"><\/td>\n<td style=\"height: 24px;\">[latex]-4.8=-4.8[\/latex]<\/td>\n<td style=\"height: 24px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]n=-6[\/latex] makes [latex]-4.8=0.8n[\/latex] a true statement, we know we have a solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146382\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146382&theme=oea&iframe_resize_id=ohm146382&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the next video to see how to solve another equation with decimals that requires division.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Solve a One Step Equation With Decimals by Dividing\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/paV-sXbRkHA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q571663\">Show Solution<\/span><\/p>\n<div id=\"q571663\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nWe will use the Multiplication Property of Equality.<\/p>\n<table id=\"eip-id1168468534925\" class=\"unnumbered unstyled\" summary=\"The first line says p divided by negative 1.8 equals negative 6.5. The next line says,\">\n<tbody>\n<tr>\n<td colspan=\"2\"><\/td>\n<td>[latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Here, <em>p<\/em> is divided by [latex]\u22121.8[\/latex]. We must multiply by [latex]\u22121.8[\/latex] to isolate <em>p<\/em><\/td>\n<td>[latex]\\color{red}{-1.8}({\\Large\\frac{p}{-1.8}})=\\color{red}{-1.8}(-6.5)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Multiply.<\/td>\n<td>[latex]p=11.7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Check:<\/strong><\/td>\n<td>[latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]{\\Large\\frac{\\color{red}{11.7}}{-1.8}}\\stackrel{?}{=}-6.5[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>\u00a0Substitute [latex]p=\\color{red}{11.7}[\/latex].<\/td>\n<td>[latex]-6.5=-6.5[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>A solution to [latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex] is [latex]p=11.7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146383\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146383&theme=oea&iframe_resize_id=ohm146383&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video shows an example of how to solve an equation with decimals that requires multiplication.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Solve a One Step Equation With Decimals by Multiplying\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/q2s9GaEwSi8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/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-3853\">\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 146375, 146377, 146379, 146382 , 146383. <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>Solving One Step Equations Using Addition and Subtraction (Decimals). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/D8wKGlxf6bM\">https:\/\/youtu.be\/D8wKGlxf6bM<\/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 Multiplying. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/q2s9GaEwSi8\">https:\/\/youtu.be\/q2s9GaEwSi8<\/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 Dividing. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/paV-sXbRkHA\">https:\/\/youtu.be\/paV-sXbRkHA<\/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":25777,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Solving One Step Equations Using Addition and Subtraction (Decimals)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/D8wKGlxf6bM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Solve a One 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GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3853","chapter","type-chapter","status-publish","hentry"],"part":5569,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3853","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3853\/revisions"}],"predecessor-version":[{"id":5584,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3853\/revisions\/5584"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/5569"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3853\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=3853"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=3853"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=3853"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=3853"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}