{"id":9359,"date":"2017-05-02T21:49:26","date_gmt":"2017-05-02T21:49:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9359"},"modified":"2019-05-24T18:05:09","modified_gmt":"2019-05-24T18:05:09","slug":"solving-equations-with-integers-using-the-addition-and-subtraction-properties-of-equality","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/chapter\/solving-equations-with-integers-using-the-addition-and-subtraction-properties-of-equality\/","title":{"raw":"Solving Equations With Integers Using Properties of Equality","rendered":"Solving Equations With Integers Using Properties of Equality"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve equations using the addition and subtraction properties of equality<\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Model the Division Property of Equality&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6529,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:4,&quot;14&quot;:[null,2,0],&quot;15&quot;:&quot;Calibri&quot;}\">Model the division property of equality<\/span><\/li>\r\n \t<li>Solve equations using the multiplication and division properties of equality<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Solve Equations with Integers Using the Addition and Subtraction Properties of Equality<\/h2>\r\nIn Solve Equations with the Subtraction and Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.\r\n<p style=\"padding-left: 60px;\">[latex]x+4=12[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]y--5=9[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">[latex]x+4\\color{red}{--4}=12\\color{red}{--4}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]y--5\\color{red}{+5}=9\\color{red}{+5}[\/latex]<\/p>\r\n<p style=\"padding-left: 60px;\">[latex]x=8[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]y=14[\/latex]<\/p>\r\nWhen you add or subtract the same quantity from both sides of an equation, you still have equality.\r\n<div class=\"textbox shaded\">\r\n<h3>Properties of Equalities<\/h3>\r\n<table id=\"fs-id1301453\" class=\"unnumbered\" summary=\"This is a table with two columns. The first column is labeled subtraction property of equality. The second column is labeled addition property of equality. In the row under the first column, subtraction property of equality, it states for any numbers, a, b, and c, if a equals b, then a minus c equals b minus c. In the row under the second column, addition property of equality, it states for any numbers a, b, and c, if a equals b, then a plus c\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Subtraction Property of Equality<\/th>\r\n<th>Addition Property of Equality<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\\text{For any numbers }a,b,c[\/latex],\r\n\r\n[latex]\\text{if }a=b\\text{ then }a-c=b-c[\/latex].<\/td>\r\n<td>[latex]\\text{For any numbers }a,b,c[\/latex],\r\n\r\n[latex]\\text{if }a=b\\text{ then }a+c=b+c[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]y+9=5[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168468293074\" class=\"unnumbered unstyled\" summary=\"This figure has 3 rows. The first row is the equation y plus 9 equals 5. The second row states subtract 9 from each side to undo the addition. It is followed by the equation y plus 9 minus 9 equals 5 minus 9. The third row states simplify. It is followed by the equation y equals negative 4\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]y+9=5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract [latex]9[\/latex] from each side to undo the addition.<\/td>\r\n<td>[latex]y+9\\color{red}{--9}=5\\color{red}{--9}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]y=--4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCheck the result by substituting [latex]-4[\/latex] into the original equation.\r\n<table id=\"eip-id1168469467685\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]y+9=5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\u22124[\/latex] for y<\/td>\r\n<td>[latex]-4+9\\stackrel{?}{=}5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]5=5\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]y=-4[\/latex] makes [latex]y+9=5[\/latex] a true statement, we found the solution to this equation.\r\n\r\n&nbsp;\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]141721[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]a - 6=-8[\/latex]\r\n[reveal-answer q=\"130879\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"130879\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468513234\" class=\"unnumbered unstyled\" summary=\"This figure has 3 rows. The first row is the equation a minus 6 equals negative 8. The second row states add 6 to each side to undo the subtraction. It is followed by the equation a minus 6 plus 6 equals negative 8 plus 6. The third row states simplify and has the equation a equals negative 2.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]a--6=--8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]6[\/latex] to each side to undo the subtraction.<\/td>\r\n<td>[latex]a--6\\color{red}{+6}=--8\\color{red}{+6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]a=--2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check the result by substituting [latex]-2[\/latex] into the original equation:<\/td>\r\n<td>[latex]a--6=--8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]-2[\/latex] for [latex]a[\/latex]<\/td>\r\n<td>[latex]--2--6\\stackrel{?}{=}--8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]--8=--8\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe solution to [latex]a - 6=-8[\/latex] is [latex]-2[\/latex].\r\nSince [latex]a=-2[\/latex] makes [latex]a - 6=-8[\/latex] a true statement, we found the 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]146557[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to solve linear equations involving integers using the addition and subtraction properties of equality.\r\n\r\nhttps:\/\/youtu.be\/xGfOlCluPDo\r\n<h2>Model the Division Property of Equality<\/h2>\r\nAll of the equations we have solved so far have been of the form [latex]x+a=b[\/latex] or [latex]x-a=b[\/latex]. We were able to isolate the variable by adding or subtracting the constant term. Now we\u2019ll see how to solve equations that involve division.\r\n\r\nWe will model an equation with envelopes and counters.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220604\/CNX_BMath_Figure_03_05_001.png\" alt=\"This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed.\" \/>\r\nHere, there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are \"hidden\" in the envelopes. So how many counters are in each envelope?\r\n\r\nTo determine the number, separate the counters on the right side into [latex]2[\/latex] groups of the same size. So [latex]6[\/latex] counters divided into [latex]2[\/latex] groups means there must be [latex]3[\/latex] counters in each group (since [latex]6\\div2=3[\/latex]).\r\n\r\nWhat equation models the situation shown in the figure below? There are two envelopes, and each contains [latex]x[\/latex] counters. Together, the two envelopes must contain a total of [latex]6[\/latex] counters. So the equation that models the situation is [latex]2x=6[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220605\/CNX_BMath_Figure_03_05_002.png\" alt=\"This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed. Under the figure is two times x equals 6.\" \/>\r\nWe can divide both sides of the equation by [latex]2[\/latex] as we did with the envelopes and counters.\r\n<p style=\"text-align: center;\">[latex]\\Large{\\frac{2x}{\\color{red}{2}}=\\frac{6}{\\color{red}{2}}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=3[\/latex]<\/p>\r\nWe found that each envelope contains [latex]\\text{3 counters.}[\/latex] Does this check? We know [latex]2\\cdot 3=6[\/latex], so it works. Three counters in each of two envelopes does equal six.\r\n\r\nAnother example is shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220607\/CNX_BMath_Figure_03_05_003.png\" alt=\"This image has two columns. In the first column are three envelopes. In the second column there are four rows of three blue circles. Underneath the image is the equation 3x equals 12.\" \/>\r\nNow we have [latex]3[\/latex] identical envelopes and [latex]\\text{12 counters.}[\/latex] How many counters are in each envelope? We have to separate the [latex]\\text{12 counters}[\/latex] into [latex]\\text{3 groups.}[\/latex] Since [latex]12\\div 3=4[\/latex], there must be [latex]\\text{4 counters}[\/latex] in each envelope.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220608\/CNX_BMath_Figure_03_05_004.png\" alt=\"This image has two columns. In the first column are four envelopes. In the second column there are twelve blue circles.\" \/>\r\nThe equation that models the situation is [latex]3x=12[\/latex]. We can divide both sides of the equation by [latex]3[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\Large{\\frac{3x}{\\color{red}{3}}=\\frac{12}{\\color{red}{3}}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=4[\/latex]<\/p>\r\nDoes this check? It does because [latex]3\\cdot 4=12[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nWrite an equation modeled by the envelopes and counters, and then solve it.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220610\/CNX_BMath_Figure_03_05_005.png\" alt=\"This image has two columns. In the first column are four envelopes. In the second column there are 8 blue circles.\" \/>\r\n\r\nSolution\r\nThere are [latex]\\text{4 envelopes,}[\/latex] or [latex]4[\/latex] unknown values, on the left that match the [latex]\\text{8 counters}[\/latex] on the right. Let\u2019s call the unknown quantity in the envelopes [latex]x[\/latex].\r\n<table id=\"eip-id1168466112781\" class=\"unnumbered unstyled\" summary=\"This figure has three rows. The first row states to write the equation. This is followed by the equation 4x equals 8. The second row states divide both sides by 4. This is followed by the equation 4x divided by 4 equals 8 divided by 4. The third row states simplify. This is followed by x equals 2.\">\r\n<tbody>\r\n<tr style=\"height: 15.46875px;\">\r\n<td style=\"height: 15.46875px;\">Write the equation.<\/td>\r\n<td style=\"height: 15.46875px;\">[latex]4x=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 41px;\">\r\n<td style=\"height: 41px;\">Divide both sides by [latex]4[\/latex].<\/td>\r\n<td style=\"height: 41px;\">[latex]\\Large{\\frac{4x}{\\color{red}{4}}=\\frac{8}{\\color{red}{4}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"height: 22px;\">Simplify.<\/td>\r\n<td style=\"height: 22px;\">[latex]x=2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThere are [latex]\\text{2 counters}[\/latex] in each envelope.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nWrite the equation modeled by the envelopes and counters. Then solve it.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220614\/CNX_BMath_Figure_03_05_006_img.png\" alt=\"This image has two columns. In the first column are four envelopes. In the second column there are 12 blue circles.\" \/>\r\n\r\n[latex]4x=12[\/latex]; [latex]x=3[\/latex]\r\n\r\nWrite the equation modeled by the envelopes and counters. Then solve it.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220616\/CNX_BMath_Figure_03_05_007_img.png\" alt=\"This image has two columns. In the first column are three envelopes. In the second column there are six blue circles.\" \/>\r\n\r\n[latex]3x=6[\/latex]; [latex]x=2[\/latex]\r\n\r\n<\/div>\r\n<h2>Solve Equations Using the Division Property of Equality<\/h2>\r\nThe previous examples lead to the Division Property of Equality. When you divide both sides of an equation by any nonzero number, you still have equality.\r\n<div class=\"textbox shaded\">\r\n<h3>Division Property of Equality<\/h3>\r\n[latex]\\begin{array}{ccc}\\text{For any numbers}&amp; a,b,c,\\text{and}&amp; c\\ne 0,\\\\ \\hfill \\text{If}&amp; a=b\\text{ then}&amp; \\large{\\frac{a}{c}=\\frac{b}{c}}.\\end{array}[\/latex]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n[latex]\\text{Solve: }7x=-49[\/latex].\r\n[reveal-answer q=\"979507\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"979507\"]\r\n\r\nSolution\r\nTo isolate [latex]x[\/latex], we need to undo multiplication.\r\n<table id=\"eip-id1168468646383\" class=\"unnumbered unstyled\" summary=\"This figure has three rows. The first row is the equation 7x equals negative 49. The second row states divide each side by 7. This is followed by the equation 7x divided by 7 equals negative 49 divided by 7. The third row states simplify and is followed by the equation x equals negative 7.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]7x=--49[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide each side by [latex]7[\/latex].<\/td>\r\n<td>[latex]\\Large{\\frac{7x}{\\color{red}{7}}=\\frac{--49}{\\color{red}{7}}}[\/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<\/tbody>\r\n<\/table>\r\nCheck the solution.\r\n<table id=\"eip-id1168469628482\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]7x=-49[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\u22127[\/latex] for x.<\/td>\r\n<td>[latex]7\\left(-7\\right)\\stackrel{?}{=}-49[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-49=-49\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTherefore, [latex]-7[\/latex] is the 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]146560[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]-3y=63[\/latex].\r\n[reveal-answer q=\"764055\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"764055\"]\r\n\r\nSolution\r\nTo isolate [latex]y[\/latex], we need to undo the multiplication.\r\n<table id=\"eip-id1168467446434\" class=\"unnumbered unstyled\" summary=\"This figure has three rows. The first row is the equation negative 3y equals 63. The second row states divide each side by negative 3. This is followed by the equation negative 3y divided by negative 3 equals 63 divided by negative 3. The third row states simplify and is followed by the equation y equals negative 21.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]--3y=63[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide each side by [latex]\u22123[\/latex].<\/td>\r\n<td>[latex]\\Large{\\frac{--3y}{\\color{red}{--3}}=\\frac{63}{\\color{red}{--3}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify<\/td>\r\n<td>[latex]y=--21[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCheck the solution.\r\n<table id=\"eip-id1168466842247\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-3y=63[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\u221221[\/latex] for y.<\/td>\r\n<td>[latex]-3\\left(-21\\right)\\stackrel{?}{=}63[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]63=63\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince this is a true statement, [latex]y=-21[\/latex] is the 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]146561[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see more examples of how to use the division and multiplication properties to solve equations with integers.\r\n\r\nhttps:\/\/youtu.be\/rZuvbYO3sV8","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve equations using the addition and subtraction properties of equality<\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Model the Division Property of Equality&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6529,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:4,&quot;14&quot;:[null,2,0],&quot;15&quot;:&quot;Calibri&quot;}\">Model the division property of equality<\/span><\/li>\n<li>Solve equations using the multiplication and division properties of equality<\/li>\n<\/ul>\n<\/div>\n<h2>Solve Equations with Integers Using the Addition and Subtraction Properties of Equality<\/h2>\n<p>In Solve Equations with the Subtraction and Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.<\/p>\n<p style=\"padding-left: 60px;\">[latex]x+4=12[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]y--5=9[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">[latex]x+4\\color{red}{--4}=12\\color{red}{--4}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]y--5\\color{red}{+5}=9\\color{red}{+5}[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">[latex]x=8[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]y=14[\/latex]<\/p>\n<p>When you add or subtract the same quantity from both sides of an equation, you still have equality.<\/p>\n<div class=\"textbox shaded\">\n<h3>Properties of Equalities<\/h3>\n<table id=\"fs-id1301453\" class=\"unnumbered\" summary=\"This is a table with two columns. The first column is labeled subtraction property of equality. The second column is labeled addition property of equality. In the row under the first column, subtraction property of equality, it states for any numbers, a, b, and c, if a equals b, then a minus c equals b minus c. In the row under the second column, addition property of equality, it states for any numbers a, b, and c, if a equals b, then a plus c\">\n<thead>\n<tr valign=\"top\">\n<th>Subtraction Property of Equality<\/th>\n<th>Addition Property of Equality<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\\text{For any numbers }a,b,c[\/latex],<\/p>\n<p>[latex]\\text{if }a=b\\text{ then }a-c=b-c[\/latex].<\/td>\n<td>[latex]\\text{For any numbers }a,b,c[\/latex],<\/p>\n<p>[latex]\\text{if }a=b\\text{ then }a+c=b+c[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]y+9=5[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468293074\" class=\"unnumbered unstyled\" summary=\"This figure has 3 rows. The first row is the equation y plus 9 equals 5. The second row states subtract 9 from each side to undo the addition. It is followed by the equation y plus 9 minus 9 equals 5 minus 9. The third row states simplify. It is followed by the equation y equals negative 4\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]y+9=5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract [latex]9[\/latex] from each side to undo the addition.<\/td>\n<td>[latex]y+9\\color{red}{--9}=5\\color{red}{--9}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]y=--4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Check the result by substituting [latex]-4[\/latex] into the original equation.<\/p>\n<table id=\"eip-id1168469467685\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]y+9=5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\u22124[\/latex] for y<\/td>\n<td>[latex]-4+9\\stackrel{?}{=}5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]5=5\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]y=-4[\/latex] makes [latex]y+9=5[\/latex] a true statement, we found the solution to this equation.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/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&theme=oea&iframe_resize_id=ohm141721&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]a - 6=-8[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q130879\">Show Solution<\/span><\/p>\n<div id=\"q130879\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468513234\" class=\"unnumbered unstyled\" summary=\"This figure has 3 rows. The first row is the equation a minus 6 equals negative 8. The second row states add 6 to each side to undo the subtraction. It is followed by the equation a minus 6 plus 6 equals negative 8 plus 6. The third row states simplify and has the equation a equals negative 2.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]a--6=--8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add [latex]6[\/latex] to each side to undo the subtraction.<\/td>\n<td>[latex]a--6\\color{red}{+6}=--8\\color{red}{+6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]a=--2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check the result by substituting [latex]-2[\/latex] into the original equation:<\/td>\n<td>[latex]a--6=--8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]-2[\/latex] for [latex]a[\/latex]<\/td>\n<td>[latex]--2--6\\stackrel{?}{=}--8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]--8=--8\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The solution to [latex]a - 6=-8[\/latex] is [latex]-2[\/latex].<br \/>\nSince [latex]a=-2[\/latex] makes [latex]a - 6=-8[\/latex] a true statement, we found the 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=\"ohm146557\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146557&theme=oea&iframe_resize_id=ohm146557&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to solve linear equations involving integers using the addition and subtraction properties of equality.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Solving One Step Equation by Add\/Subtracting Integers (Var on Left)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/xGfOlCluPDo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Model the Division Property of Equality<\/h2>\n<p>All of the equations we have solved so far have been of the form [latex]x+a=b[\/latex] or [latex]x-a=b[\/latex]. We were able to isolate the variable by adding or subtracting the constant term. Now we\u2019ll see how to solve equations that involve division.<\/p>\n<p>We will model an equation with envelopes and counters.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220604\/CNX_BMath_Figure_03_05_001.png\" alt=\"This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed.\" \/><br \/>\nHere, there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are &#8220;hidden&#8221; in the envelopes. So how many counters are in each envelope?<\/p>\n<p>To determine the number, separate the counters on the right side into [latex]2[\/latex] groups of the same size. So [latex]6[\/latex] counters divided into [latex]2[\/latex] groups means there must be [latex]3[\/latex] counters in each group (since [latex]6\\div2=3[\/latex]).<\/p>\n<p>What equation models the situation shown in the figure below? There are two envelopes, and each contains [latex]x[\/latex] counters. Together, the two envelopes must contain a total of [latex]6[\/latex] counters. So the equation that models the situation is [latex]2x=6[\/latex].<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220605\/CNX_BMath_Figure_03_05_002.png\" alt=\"This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed. Under the figure is two times x equals 6.\" \/><br \/>\nWe can divide both sides of the equation by [latex]2[\/latex] as we did with the envelopes and counters.<\/p>\n<p style=\"text-align: center;\">[latex]\\Large{\\frac{2x}{\\color{red}{2}}=\\frac{6}{\\color{red}{2}}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=3[\/latex]<\/p>\n<p>We found that each envelope contains [latex]\\text{3 counters.}[\/latex] Does this check? We know [latex]2\\cdot 3=6[\/latex], so it works. Three counters in each of two envelopes does equal six.<\/p>\n<p>Another example is shown below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220607\/CNX_BMath_Figure_03_05_003.png\" alt=\"This image has two columns. In the first column are three envelopes. In the second column there are four rows of three blue circles. Underneath the image is the equation 3x equals 12.\" \/><br \/>\nNow we have [latex]3[\/latex] identical envelopes and [latex]\\text{12 counters.}[\/latex] How many counters are in each envelope? We have to separate the [latex]\\text{12 counters}[\/latex] into [latex]\\text{3 groups.}[\/latex] Since [latex]12\\div 3=4[\/latex], there must be [latex]\\text{4 counters}[\/latex] in each envelope.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220608\/CNX_BMath_Figure_03_05_004.png\" alt=\"This image has two columns. In the first column are four envelopes. In the second column there are twelve blue circles.\" \/><br \/>\nThe equation that models the situation is [latex]3x=12[\/latex]. We can divide both sides of the equation by [latex]3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\Large{\\frac{3x}{\\color{red}{3}}=\\frac{12}{\\color{red}{3}}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=4[\/latex]<\/p>\n<p>Does this check? It does because [latex]3\\cdot 4=12[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Write an equation modeled by the envelopes and counters, and then solve it.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220610\/CNX_BMath_Figure_03_05_005.png\" alt=\"This image has two columns. In the first column are four envelopes. In the second column there are 8 blue circles.\" \/><\/p>\n<p>Solution<br \/>\nThere are [latex]\\text{4 envelopes,}[\/latex] or [latex]4[\/latex] unknown values, on the left that match the [latex]\\text{8 counters}[\/latex] on the right. Let\u2019s call the unknown quantity in the envelopes [latex]x[\/latex].<\/p>\n<table id=\"eip-id1168466112781\" class=\"unnumbered unstyled\" summary=\"This figure has three rows. The first row states to write the equation. This is followed by the equation 4x equals 8. The second row states divide both sides by 4. This is followed by the equation 4x divided by 4 equals 8 divided by 4. The third row states simplify. This is followed by x equals 2.\">\n<tbody>\n<tr style=\"height: 15.46875px;\">\n<td style=\"height: 15.46875px;\">Write the equation.<\/td>\n<td style=\"height: 15.46875px;\">[latex]4x=8[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 41px;\">\n<td style=\"height: 41px;\">Divide both sides by [latex]4[\/latex].<\/td>\n<td style=\"height: 41px;\">[latex]\\Large{\\frac{4x}{\\color{red}{4}}=\\frac{8}{\\color{red}{4}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"height: 22px;\">Simplify.<\/td>\n<td style=\"height: 22px;\">[latex]x=2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>There are [latex]\\text{2 counters}[\/latex] in each envelope.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Write the equation modeled by the envelopes and counters. Then solve it.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220614\/CNX_BMath_Figure_03_05_006_img.png\" alt=\"This image has two columns. In the first column are four envelopes. In the second column there are 12 blue circles.\" \/><\/p>\n<p>[latex]4x=12[\/latex]; [latex]x=3[\/latex]<\/p>\n<p>Write the equation modeled by the envelopes and counters. Then solve it.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220616\/CNX_BMath_Figure_03_05_007_img.png\" alt=\"This image has two columns. In the first column are three envelopes. In the second column there are six blue circles.\" \/><\/p>\n<p>[latex]3x=6[\/latex]; [latex]x=2[\/latex]<\/p>\n<\/div>\n<h2>Solve Equations Using the Division Property of Equality<\/h2>\n<p>The previous examples lead to the Division Property of Equality. When you divide both sides of an equation by any nonzero number, you still have equality.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division Property of Equality<\/h3>\n<p>[latex]\\begin{array}{ccc}\\text{For any numbers}& a,b,c,\\text{and}& c\\ne 0,\\\\ \\hfill \\text{If}& a=b\\text{ then}& \\large{\\frac{a}{c}=\\frac{b}{c}}.\\end{array}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>[latex]\\text{Solve: }7x=-49[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q979507\">Show Solution<\/span><\/p>\n<div id=\"q979507\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo isolate [latex]x[\/latex], we need to undo multiplication.<\/p>\n<table id=\"eip-id1168468646383\" class=\"unnumbered unstyled\" summary=\"This figure has three rows. The first row is the equation 7x equals negative 49. The second row states divide each side by 7. This is followed by the equation 7x divided by 7 equals negative 49 divided by 7. The third row states simplify and is followed by the equation x equals negative 7.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]7x=--49[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide each side by [latex]7[\/latex].<\/td>\n<td>[latex]\\Large{\\frac{7x}{\\color{red}{7}}=\\frac{--49}{\\color{red}{7}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x=--7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Check the solution.<\/p>\n<table id=\"eip-id1168469628482\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]7x=-49[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\u22127[\/latex] for x.<\/td>\n<td>[latex]7\\left(-7\\right)\\stackrel{?}{=}-49[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-49=-49\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Therefore, [latex]-7[\/latex] is the 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=\"ohm146560\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146560&theme=oea&iframe_resize_id=ohm146560&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]-3y=63[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q764055\">Show Solution<\/span><\/p>\n<div id=\"q764055\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo isolate [latex]y[\/latex], we need to undo the multiplication.<\/p>\n<table id=\"eip-id1168467446434\" class=\"unnumbered unstyled\" summary=\"This figure has three rows. The first row is the equation negative 3y equals 63. The second row states divide each side by negative 3. This is followed by the equation negative 3y divided by negative 3 equals 63 divided by negative 3. The third row states simplify and is followed by the equation y equals negative 21.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]--3y=63[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide each side by [latex]\u22123[\/latex].<\/td>\n<td>[latex]\\Large{\\frac{--3y}{\\color{red}{--3}}=\\frac{63}{\\color{red}{--3}}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify<\/td>\n<td>[latex]y=--21[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Check the solution.<\/p>\n<table id=\"eip-id1168466842247\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-3y=63[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\u221221[\/latex] for y.<\/td>\n<td>[latex]-3\\left(-21\\right)\\stackrel{?}{=}63[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]63=63\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since this is a true statement, [latex]y=-21[\/latex] is the 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=\"ohm146561\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146561&theme=oea&iframe_resize_id=ohm146561&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see more examples of how to use the division and multiplication properties to solve equations with integers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Solving One Step Equation by Mult\/Div.  Integers (Var on Left)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/rZuvbYO3sV8?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-9359\">\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>Queston ID: 146560, 146561. <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: Solving One Step Equation by Add\/Subtracting Integers (Var on Left). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/xGfOlCluPDo\">https:\/\/youtu.be\/xGfOlCluPDo<\/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><strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Provided by<\/strong>: Ex: Solving One Step Equation by Mult\/Div. Integers (Var on Left). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"\"><\/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":17533,"menu_order":26,"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\":\"Ex: Solving One Step Equation by Add\/Subtracting Integers (Var on Left)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/xGfOlCluPDo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"Ex: Solving One Step Equation by Mult\/Div. Integers (Var on Left)\",\"url\":\"xhttps:\/\/youtu.be\/rZuvbYO3sV8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Queston ID: 146560, 146561\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"99af8f2d-2cd4-45d9-8db7-232d3918d8d4","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-9359","chapter","type-chapter","status-publish","hentry"],"part":6350,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9359","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":30,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9359\/revisions"}],"predecessor-version":[{"id":15827,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9359\/revisions\/15827"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/parts\/6350"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapters\/9359\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/media?parent=9359"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=9359"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/contributor?post=9359"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/wp-json\/wp\/v2\/license?post=9359"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}