{"id":496,"date":"2025-07-10T17:45:15","date_gmt":"2025-07-10T17:45:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=496"},"modified":"2026-01-07T22:27:38","modified_gmt":"2026-01-07T22:27:38","slug":"working-with-functions-background-youll-need-1-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/working-with-functions-background-youll-need-1-2\/","title":{"raw":"Working with Functions: Background You'll Need 1","rendered":"Working with Functions: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Simplify and evaluate an algebraic expression.<\/span><\/li>\r\n<\/ul>\r\n<\/section>An <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. For example, [latex]3x + 2y - 7[\/latex] is an algebraic expression that contains two variables [latex]x[\/latex] and [latex]y[\/latex] and three constants [latex]3[\/latex], [latex]2[\/latex], and [latex]7[\/latex].\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>constant, variable, algebraic expression<\/h3>\r\n<ul>\r\n \t<li>A <strong>constant<\/strong> is a fixed value or a number that does not change in a particular context.<\/li>\r\n \t<li>A <strong>variable<\/strong> is a symbol that represents a value or quantity that can change or vary in a given situation or context.<\/li>\r\n \t<li>An <strong>algebraic expression<\/strong> is a mathematical phrase or combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression.\r\n\r\n<section class=\"textbox questionHelp\"><strong>How To: Evaluate Algebraic Expressions<\/strong>Use the following steps to evaluate an algebraic expression:\r\n<ol>\r\n \t<li>Replace each variable in the expression with the given value<\/li>\r\n \t<li>Simplify the resulting expression using the order of operations<\/li>\r\n<\/ol>\r\nNote: If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Evaluate each expression for the given values.\r\n<ol>\r\n \t<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\r\n \t<li>[latex]\\frac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{3}\\pi {r}^{3}[\/latex] for [latex]r=5[\/latex]<\/li>\r\n \t<li>[latex]a+ab+b[\/latex] for [latex]a=11,b=-8[\/latex]<\/li>\r\n \t<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex] for [latex]m=2,n=3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"182854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"182854\"]\r\n<ol>\r\n \t<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}x+5 &amp;=\\left(-5\\right)+5 \\\\ &amp;=0\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]10[\/latex] for [latex]t[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{t}{2t-1} &amp; =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ &amp; =\\frac{10}{20-1} \\\\ &amp; =\\frac{10}{19}\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]5[\/latex] for [latex]r[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{3}\\pi r^{3} &amp; =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ &amp; =\\frac{4}{3}\\pi\\left(125\\right) \\\\ &amp; =\\frac{500}{3}\\pi\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]11[\/latex] for [latex]a[\/latex] and [latex]\u20138[\/latex] for [latex]b[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}a+ab+b &amp; =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ &amp; =11-8-8 \\\\ &amp; =-85\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]2[\/latex] for [latex]m[\/latex] and [latex]3[\/latex] for [latex]n[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{2m^{3}n^{2}} &amp; =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ &amp; =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ &amp; =\\sqrt{144} \\\\ &amp; =12\\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Formulas<\/h2>\r\nAn <strong>equation<\/strong> is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[\/latex] has the unique solution [latex]x=3[\/latex] because when we substitute [latex]3[\/latex] for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2\\left(3\\right)+1=7[\/latex].\r\n\r\nA <strong>formula<\/strong> is an equation expressing a relationship between constant and variable quantities. Very often the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[\/latex] of a circle in terms of the radius [latex]r[\/latex] of the circle: [latex]A=\\pi {r}^{2}[\/latex]. For any value of [latex]r[\/latex], the area [latex]A[\/latex] can be found by evaluating the expression [latex]\\pi {r}^{2}[\/latex].\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>Equations and formulas<\/h3>\r\n<ul>\r\n \t<li>An <strong>equation<\/strong> is a mathematical statement that shows the equality of two expressions, typically separated by an equal sign. It states that the two expressions have the same value, and the values of variables that make the equation true are called solutions.<\/li>\r\n \t<li>A <strong>formula<\/strong> is a mathematical expression that represents a relationship or a rule between variables or quantities. It usually contains variables, constants, and arithmetic operations, and is used to calculate or derive a particular result or value.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section class=\"textbox example\">A right circular cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex] has the surface area [latex]S[\/latex] (in square units) given by the formula [latex]S=2\\pi r\\left(r+h\\right)[\/latex].\r\n[latex]\\\\[\/latex]\r\nFind the surface area of a cylinder with radius [latex]6[\/latex] in. and height [latex]9[\/latex] in. <em>Leave the answer in terms of [latex]\\pi[\/latex].<\/em><center><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223817\/CNX_CAT_Figure_01_01_004.jpg\" alt=\"A right circular cylinder with an arrow extending from the center of the top circle outward to the edge, labeled: r. Another arrow beside the image going from top to bottom, labeled: h.\" width=\"487\" height=\"279\" \/><\/center><center><strong><span style=\"font-size: 10pt;\">Right circular cylinder<\/span><\/strong><\/center>[reveal-answer q=\"257174\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"257174\"]Evaluate the expression [latex]2\\pi r\\left(r+h\\right)[\/latex] for [latex]r=6[\/latex] and [latex]h=9[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align}S&amp;=2\\pi r\\left(r+h\\right) \\\\ &amp; =2\\pi\\left(6\\right)[\\left(6\\right)+\\left(9\\right)] \\\\ &amp; =2\\pi\\left(6\\right)\\left(15\\right) \\\\ &amp; =180\\pi\\end{align}[\/latex]<\/p>\r\nThe surface area is [latex]180\\pi [\/latex] square inches.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox seeExample\"><center>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223819\/CNX_CAT_Figure_01_01_005.jpg\" alt=\"An art frame with a piece of artwork in the center. The frame has a width of 8 centimeters. The artwork itself has a length of 32 centimeters and a width of 24 centimeters.\" width=\"487\" height=\"407\" \/> Framed artwork with dimensions[\/caption]\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/center>A photograph with length [latex]L[\/latex] and width [latex]W[\/latex] is placed in a mat of width [latex]8[\/latex] centimeters (cm). The area of the mat (in square centimeters, or cm<sup>[latex]2[\/latex]<\/sup>) is found to be [latex]A=\\left(L+16\\right)\\left(W+16\\right)-L\\cdot W[\/latex]. Find the area of a mat for a photograph with length [latex]32[\/latex] cm and width [latex]24[\/latex] cm.[reveal-answer q=\"846181\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"846181\"][latex]1,152 cm^{2}[\/latex][\/hidden-answer]<\/section>\r\n<h2>Simplify Algebraic Expressions<\/h2>\r\nSometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.\r\n\r\n<section class=\"textbox example\">Simplify the following algebraic expressions:\r\n<ol>\r\n \t<li>[latex]3x - 2y+x - 3y - 7[\/latex]<\/li>\r\n \t<li>[latex]2r - 5\\left(3-r\\right)+4[\/latex]<\/li>\r\n \t<li>[latex]\\left(4t-\\dfrac{5}{4}s\\right)-\\left(\\dfrac{2}{3}t+2s\\right)[\/latex]<\/li>\r\n \t<li>[latex]2mn - 5m+3mn+n[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"286046\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"286046\"]\r\n<ol>\r\n \t<li><center>[latex]\\begin{align}3x-2y+x-3y-7 &amp; =3x+x-2y-3y-7 &amp;&amp; \\text{Commutative property of addition} \\\\ &amp; =4x-5y-7 &amp;&amp; \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/center><\/li>\r\n \t<li><center>[latex]\\begin{align}2r-5\\left(3-r\\right)+4 &amp; =2r-15+5r+4 &amp;&amp; \\text{Distributive property}\\\\&amp;=2r+5r-15+4 &amp;&amp; \\text{Commutative property of addition} \\\\ &amp; =7r-11 &amp;&amp; \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/center><\/li>\r\n \t<li><center>[latex]\\begin{align} 4t-\\frac{5}{4}s -\\left(\\frac{2}{3}t+2s\\right) &amp;=4t-\\frac{5}{4}s-\\frac{2}{3}t-2s &amp;&amp;\\text{Distributive property}\\\\&amp;=4t-\\frac{2}{3}t-\\frac{5}{4}s-2s &amp;&amp; \\text{Commutative property of addition}\\\\&amp;=\\frac{12}{3}t-\\frac{2}{3}t-\\frac{5}{4}s-\\frac{8}{4}s &amp;&amp; \\text{Common Denominators}\\\\ &amp; =\\frac{10}{3}t-\\frac{13}{4}s &amp;&amp; \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/center><\/li>\r\n \t<li><center>[latex]\\begin{align}mn-5m+3mn+n &amp; =2mn+3mn-5m+n &amp;&amp; \\text{Commutative property of addition} \\\\ &amp; =5mn-5m+n &amp;&amp; \\text{Simplify}\\end{align}[\/latex]<\/center><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317889[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\">Simplify and evaluate an algebraic expression.<\/span><\/li>\n<\/ul>\n<\/section>\n<p>An <strong>algebraic expression<\/strong> is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. For example, [latex]3x + 2y - 7[\/latex] is an algebraic expression that contains two variables [latex]x[\/latex] and [latex]y[\/latex] and three constants [latex]3[\/latex], [latex]2[\/latex], and [latex]7[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>constant, variable, algebraic expression<\/h3>\n<ul>\n<li>A <strong>constant<\/strong> is a fixed value or a number that does not change in a particular context.<\/li>\n<li>A <strong>variable<\/strong> is a symbol that represents a value or quantity that can change or vary in a given situation or context.<\/li>\n<li>An <strong>algebraic expression<\/strong> is a mathematical phrase or combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<p>Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression.<\/p>\n<section class=\"textbox questionHelp\"><strong>How To: Evaluate Algebraic Expressions<\/strong>Use the following steps to evaluate an algebraic expression:<\/p>\n<ol>\n<li>Replace each variable in the expression with the given value<\/li>\n<li>Simplify the resulting expression using the order of operations<\/li>\n<\/ol>\n<p>Note: If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate each expression for the given values.<\/p>\n<ol>\n<li>[latex]x+5[\/latex] for [latex]x=-5[\/latex]<\/li>\n<li>[latex]\\frac{t}{2t - 1}[\/latex] for [latex]t=10[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{3}\\pi {r}^{3}[\/latex] for [latex]r=5[\/latex]<\/li>\n<li>[latex]a+ab+b[\/latex] for [latex]a=11,b=-8[\/latex]<\/li>\n<li>[latex]\\sqrt{2{m}^{3}{n}^{2}}[\/latex] for [latex]m=2,n=3[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q182854\">Show Solution<\/button><\/p>\n<div id=\"q182854\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Substitute [latex]-5[\/latex] for [latex]x[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}x+5 &=\\left(-5\\right)+5 \\\\ &=0\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]10[\/latex] for [latex]t[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{t}{2t-1} & =\\frac{\\left(10\\right)}{2\\left(10\\right)-1} \\\\ & =\\frac{10}{20-1} \\\\ & =\\frac{10}{19}\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]5[\/latex] for [latex]r[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\frac{4}{3}\\pi r^{3} & =\\frac{4}{3}\\pi\\left(5\\right)^{3} \\\\ & =\\frac{4}{3}\\pi\\left(125\\right) \\\\ & =\\frac{500}{3}\\pi\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]11[\/latex] for [latex]a[\/latex] and [latex]\u20138[\/latex] for [latex]b[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}a+ab+b & =\\left(11\\right)+\\left(11\\right)\\left(-8\\right)+\\left(-8\\right) \\\\ & =11-8-8 \\\\ & =-85\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]2[\/latex] for [latex]m[\/latex] and [latex]3[\/latex] for [latex]n[\/latex].\n<div style=\"text-align: center;\">[latex]\\begin{align}\\sqrt{2m^{3}n^{2}} & =\\sqrt{2\\left(2\\right)^{3}\\left(3\\right)^{2}} \\\\ & =\\sqrt{2\\left(8\\right)\\left(9\\right)} \\\\ & =\\sqrt{144} \\\\ & =12\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Formulas<\/h2>\n<p>An <strong>equation<\/strong> is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation [latex]2x+1=7[\/latex] has the unique solution [latex]x=3[\/latex] because when we substitute [latex]3[\/latex] for [latex]x[\/latex] in the equation, we obtain the true statement [latex]2\\left(3\\right)+1=7[\/latex].<\/p>\n<p>A <strong>formula<\/strong> is an equation expressing a relationship between constant and variable quantities. Very often the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area [latex]A[\/latex] of a circle in terms of the radius [latex]r[\/latex] of the circle: [latex]A=\\pi {r}^{2}[\/latex]. For any value of [latex]r[\/latex], the area [latex]A[\/latex] can be found by evaluating the expression [latex]\\pi {r}^{2}[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>Equations and formulas<\/h3>\n<ul>\n<li>An <strong>equation<\/strong> is a mathematical statement that shows the equality of two expressions, typically separated by an equal sign. It states that the two expressions have the same value, and the values of variables that make the equation true are called solutions.<\/li>\n<li>A <strong>formula<\/strong> is a mathematical expression that represents a relationship or a rule between variables or quantities. It usually contains variables, constants, and arithmetic operations, and is used to calculate or derive a particular result or value.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox example\">A right circular cylinder with radius [latex]r[\/latex] and height [latex]h[\/latex] has the surface area [latex]S[\/latex] (in square units) given by the formula [latex]S=2\\pi r\\left(r+h\\right)[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\nFind the surface area of a cylinder with radius [latex]6[\/latex] in. and height [latex]9[\/latex] in. <em>Leave the answer in terms of [latex]\\pi[\/latex].<\/em><\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223817\/CNX_CAT_Figure_01_01_004.jpg\" alt=\"A right circular cylinder with an arrow extending from the center of the top circle outward to the edge, labeled: r. Another arrow beside the image going from top to bottom, labeled: h.\" width=\"487\" height=\"279\" \/><\/div>\n<div style=\"text-align: center;\"><strong><span style=\"font-size: 10pt;\">Right circular cylinder<\/span><\/strong><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q257174\">Show Solution<\/button><\/p>\n<div id=\"q257174\" class=\"hidden-answer\" style=\"display: none\">Evaluate the expression [latex]2\\pi r\\left(r+h\\right)[\/latex] for [latex]r=6[\/latex] and [latex]h=9[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}S&=2\\pi r\\left(r+h\\right) \\\\ & =2\\pi\\left(6\\right)[\\left(6\\right)+\\left(9\\right)] \\\\ & =2\\pi\\left(6\\right)\\left(15\\right) \\\\ & =180\\pi\\end{align}[\/latex]<\/p>\n<p>The surface area is [latex]180\\pi[\/latex] square inches.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox seeExample\">\n<div style=\"text-align: center;\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223819\/CNX_CAT_Figure_01_01_005.jpg\" alt=\"An art frame with a piece of artwork in the center. The frame has a width of 8 centimeters. The artwork itself has a length of 32 centimeters and a width of 24 centimeters.\" width=\"487\" height=\"407\" \/><figcaption class=\"wp-caption-text\">Framed artwork with dimensions<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>A photograph with length [latex]L[\/latex] and width [latex]W[\/latex] is placed in a mat of width [latex]8[\/latex] centimeters (cm). The area of the mat (in square centimeters, or cm<sup>[latex]2[\/latex]<\/sup>) is found to be [latex]A=\\left(L+16\\right)\\left(W+16\\right)-L\\cdot W[\/latex]. Find the area of a mat for a photograph with length [latex]32[\/latex] cm and width [latex]24[\/latex] cm.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q846181\">Show Solution<\/button><\/p>\n<div id=\"q846181\" class=\"hidden-answer\" style=\"display: none\">[latex]1,152 cm^{2}[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Simplify Algebraic Expressions<\/h2>\n<p>Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.<\/p>\n<section class=\"textbox example\">Simplify the following algebraic expressions:<\/p>\n<ol>\n<li>[latex]3x - 2y+x - 3y - 7[\/latex]<\/li>\n<li>[latex]2r - 5\\left(3-r\\right)+4[\/latex]<\/li>\n<li>[latex]\\left(4t-\\dfrac{5}{4}s\\right)-\\left(\\dfrac{2}{3}t+2s\\right)[\/latex]<\/li>\n<li>[latex]2mn - 5m+3mn+n[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q286046\">Show Solution<\/button><\/p>\n<div id=\"q286046\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<div style=\"text-align: center;\">[latex]\\begin{align}3x-2y+x-3y-7 & =3x+x-2y-3y-7 && \\text{Commutative property of addition} \\\\ & =4x-5y-7 && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<\/li>\n<li>\n<div style=\"text-align: center;\">[latex]\\begin{align}2r-5\\left(3-r\\right)+4 & =2r-15+5r+4 && \\text{Distributive property}\\\\&=2r+5r-15+4 && \\text{Commutative property of addition} \\\\ & =7r-11 && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<\/li>\n<li>\n<div style=\"text-align: center;\">[latex]\\begin{align} 4t-\\frac{5}{4}s -\\left(\\frac{2}{3}t+2s\\right) &=4t-\\frac{5}{4}s-\\frac{2}{3}t-2s &&\\text{Distributive property}\\\\&=4t-\\frac{2}{3}t-\\frac{5}{4}s-2s && \\text{Commutative property of addition}\\\\&=\\frac{12}{3}t-\\frac{2}{3}t-\\frac{5}{4}s-\\frac{8}{4}s && \\text{Common Denominators}\\\\ & =\\frac{10}{3}t-\\frac{13}{4}s && \\text{Simplify} \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<\/li>\n<li>\n<div style=\"text-align: center;\">[latex]\\begin{align}mn-5m+3mn+n & =2mn+3mn-5m+n && \\text{Commutative property of addition} \\\\ & =5mn-5m+n && \\text{Simplify}\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm317889\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317889&theme=lumen&iframe_resize_id=ohm317889&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":498,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/496"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/496\/revisions"}],"predecessor-version":[{"id":5226,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/496\/revisions\/5226"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/498"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/496\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=496"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=496"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=496"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=496"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}