{"id":2830,"date":"2024-02-09T19:26:06","date_gmt":"2024-02-09T19:26:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=2830"},"modified":"2024-02-09T19:26:11","modified_gmt":"2024-02-09T19:26:11","slug":"5-2-1-simplifying-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/5-2-1-simplifying-expressions\/","title":{"raw":"5.2.1 Simplifying Expressions","rendered":"5.2.1 Simplifying Expressions"},"content":{"raw":"<!-- Le HTML5 shim, for IE6-8 support of HTML5 elements --><!-- [if lt IE 9]>\r\n<script src=\"https:\/\/html5shim.googlecode.com\/svn\/trunk\/html5.js\">\r\n\t<\/script>\r\n<![endif]-->\r\n<div class=\"wrapper\">\r\n<div id=\"wrap\">\r\n<div id=\"content\" role=\"main\">\r\n<div id=\"post-67\" class=\"standard post-67 chapter type-chapter status-publish hentry\">\r\n<div class=\"entry-content\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the coefficient of a variable term<\/li>\r\n \t<li>Recognize and combine like terms in an expression<\/li>\r\n \t<li>Use the order of operations to simplify expressions containing like terms<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li><strong>Term<\/strong>: a constant or the product or quotient of constants and one or more variables<\/li>\r\n \t<li><strong>Constant<\/strong>: a number that has a fixed value<\/li>\r\n \t<li><strong>Variable<\/strong>: a letter that represents a value that can change<\/li>\r\n \t<li><strong>Algebraic expression<\/strong>:\u00a0is a term, or the sum or difference of terms<\/li>\r\n \t<li><strong>Coefficient<\/strong>: the number in front of a variable or term<\/li>\r\n \t<li><strong>Like<\/strong> <strong>terms<\/strong>: terms that have the same variables and exponents<\/li>\r\n \t<li><strong>Commutative Property of Addition<\/strong>:\u00a0we can change the order of addends without changing the sum<\/li>\r\n \t<li><strong>Associative Property of Addition<\/strong>:\u00a0we can regroup the addends without changing the sum<\/li>\r\n \t<li><strong>Commutative Property of Multiplication<\/strong>: we can change the order of the factors without changing the product<\/li>\r\n \t<li><strong> Associative Property of Multiplication<\/strong>:\u00a0we can regroup the factors without changing the product<\/li>\r\n \t<li><strong>Distributive Property of Multiplication over Addition<\/strong>: a constant is multiplied onto an expression by multiplying each term in the expression by the constant<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 data-type=\"title\">Identify Terms, Coefficients, and Like Terms<\/h2>\r\nIn mathematics, we may see expressions such as [latex]x+5[\/latex],\u00a0 [latex]\\dfrac{4}{3}{r}^{3}[\/latex], or [latex]5m-2n+6mn[\/latex]. Algebraic expressions are made up of\u00a0<strong>terms<\/strong>.\r\n\r\nA <em><strong>term<\/strong><\/em> is a constant or the product or quotient of constants and one or more variables. Some examples of terms are [latex]7,y,5{x}^{2},-\\frac{9a}{2b},\\text{ and }-13xy[\/latex].\r\n\r\nIn the expression [latex]x+5[\/latex], [latex]5[\/latex] is called a\u00a0<strong>constant\u00a0<\/strong>because it does not vary and\u00a0[latex]x[\/latex]\u00a0is called a\u00a0<strong>variable\u00a0<\/strong>because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An\u00a0<strong><em>algebraic expression\u00a0<\/em><\/strong>is a term, or the sum of terms (remember that subtraction can always be written as a sum of the opposite).\r\n\r\nThe constant that multiplies the variable(s) in a term is called the\u00a0<strong>coefficient<\/strong>. We can think of the coefficient as the number\u00a0<em data-effect=\"italics\">in front of\u00a0<\/em>the variable. The coefficient of the term [latex]3x[\/latex] is [latex]3[\/latex]. When we write [latex]x[\/latex], the coefficient is [latex]1[\/latex], since [latex]x=1\\cdot x[\/latex]. The table below gives the coefficients for each of the terms in the left column.\r\n<table id=\"fs-id2266631\" summary=\"This table has five rows and two columns. The first row is a header row and it labels each column. The first column is labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"center\">Term<\/th>\r\n<th data-align=\"center\">Coefficient<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]7[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]-9a[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]y[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]-\\frac{2}{5}{x}^{2}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]-\\frac{2}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation with the term when we list it. Think of the operation as belonging to the term it precedes.\r\n<table id=\"fs-id1596496\" summary=\"This table has six rows and two columns. The first row is a header row and it labels each column. The first column is labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"center\">Expression<\/th>\r\n<th data-align=\"center\">Terms<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]-7[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]y[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]y[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]x-\\frac{7}{5}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]x,-\\frac{7}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]2x+7y-4[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]2x,7y,-4[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]3{x}^{2}+4{x}^{2}+5y+3[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]3{x}^{2},4{x}^{2},5y,3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div>Notice that when a term is being subtracted, the coefficient and the term are negative.<\/div>\r\n<div><\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIdentify each term in the expression [latex]9b+15{x}^{2}-a+6[\/latex]. Then identify the coefficient of each term.\r\n\r\nSolution:\r\n\r\nThe expression has four terms. They are [latex]9b,15{x}^{2},-a[\/latex], and [latex]6[\/latex].\r\n<ul>\r\n \t<li>The coefficient of [latex]9b[\/latex] is [latex]9[\/latex].<\/li>\r\n \t<li>The coefficient of [latex]15{x}^{2}[\/latex] is [latex]15[\/latex].<\/li>\r\n \t<li>The coefficient of [latex]a[\/latex] is [latex]-1[\/latex].<\/li>\r\n \t<li>The coefficient of a constant is the constant, so the coefficient of [latex]6[\/latex] is [latex]6[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"ohm144899\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144899&amp;theme=oea&amp;iframe_resize_id=ohm144899&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\nWhat exactly does it mean for a constant and a variable to be multiplied or divided? Let's consider [latex]4x[\/latex]. Multiplication is repeated addition, so\u00a0[latex]4x=x+x+x+x[\/latex].\u00a0 What about [latex]\\frac{1}{5}x[\/latex]?\u00a0 Multiplying by\u00a0[latex]\\frac{1}{5}[\/latex] is equivalent to dividing by [latex]5[\/latex]. So,\u00a0[latex]\\frac{1}{5}x=\\frac{x}{5}[\/latex]. In other words, we take the quantity\u00a0[latex]x[\/latex] and divide it by[latex]5[\/latex].\r\n\r\nRecall that exponents are a more efficient way to write repeated multiplication. So, a variable with an exponent means to multiply the variable the exponent number of times. For example, [latex]x^3=x\\cdot x\\cdot x[\/latex] and [latex]y^5=y\\cdot y\\cdot y\\cdot y\\cdot y[\/latex].\r\n\r\nSome terms share common traits. Look at the following terms. Which ones seem to have traits in common?\r\n\r\n[latex]5x,7,{n}^{2},4,-3x,9{n}^{2}[\/latex]\r\n<ul id=\"fs-id1627987\" data-bullet-style=\"bullet\">\r\n \t<li>The terms [latex]7[\/latex] and [latex]4[\/latex] are both constant terms.<\/li>\r\n \t<li>The terms [latex]5x[\/latex] and [latex]-3x[\/latex] are both terms with [latex]x[\/latex].<\/li>\r\n \t<li>The terms [latex]{n}^{2}[\/latex] and [latex]9{n}^{2}[\/latex] both have [latex]{n}^{2}[\/latex].<\/li>\r\n<\/ul>\r\nTerms are called <em><strong>like<\/strong> <strong>terms<\/strong><\/em> if they have the same variables and exponents. All constant terms are also like terms. So among the terms [latex]5x,7,{n}^{2},4,-3x,9{n}^{2}[\/latex],\r\n<ul>\r\n \t<li>[latex]7[\/latex] and [latex]4[\/latex] are like terms.<\/li>\r\n \t<li>[latex]5x[\/latex] and [latex]3-x[\/latex] are like terms.<\/li>\r\n \t<li>[latex]{n}^{2}[\/latex] and [latex]9{n}^{2}[\/latex] are like terms.<\/li>\r\n<\/ul>\r\n<div class=\"textbox shaded\">\r\n<h3>Like Terms<\/h3>\r\nTerms that are either constants or have the same variables with the same exponents are <strong>like terms<\/strong>.\r\n\r\n<\/div>\r\nLike terms\u00a0are terms where the variables match exactly (exponents included). Examples of like terms would be [latex]5xy[\/latex] and [latex]-3xy[\/latex], or [latex]8a^2b[\/latex] and [latex]a^2b[\/latex], or [latex]-\\frac{3}{4}[\/latex] and [latex]8[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIdentify the like terms:\r\n<ol>\r\n \t<li>[latex]{y}^{3},7{x}^{2},-14,\\frac{2}{3},4{y}^{3},-9x,5{x}^{2}[\/latex]<\/li>\r\n \t<li>[latex]-4{x}^{2}+2x+5{x}^{2}+6x-40x+8xy[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution:<\/h4>\r\n1. [latex]{y}^{3},7{x}^{2},-14,\\frac{2}{3},-4{y}^{3},-9x,5{x}^{2}[\/latex]\r\n\r\nLook at the variables and exponents. The expression contains [latex]{y}^{3},{x}^{2},x[\/latex], and constants.\r\n\r\nThe terms [latex]{y}^{3}[\/latex] and [latex]-4{y}^{3}[\/latex] are like terms because they both contain [latex]{y}^{3}[\/latex].\r\n\r\nThe terms [latex]7{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both contain [latex]{x}^{2}[\/latex].\r\n\r\nThe terms [latex]-14[\/latex] and [latex]\\frac{2}{3}[\/latex] are like terms because they are both constants.\r\n\r\nThe term [latex]-9x[\/latex] does not have any like terms in this list since no other terms have the variable [latex]x[\/latex] raised to the power of [latex]1[\/latex].\r\n\r\n2. [latex]-4{x}^{2}+2x+5{x}^{2}+6x-40x+8xy[\/latex]\r\n\r\nLook at the variables and exponents. The expression contains [latex]{x}^{2},x,text{and}xy[\/latex] terms.\r\n\r\nThe terms [latex]-4{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both contain [latex]{x}^{2}[\/latex].\r\n\r\nThe terms [latex]2x,6x,\\text{and}-40x[\/latex] are like terms because they all contain [latex]x[\/latex].\r\n\r\nThe term [latex]8xy[\/latex] has no like terms in the given expression because no other terms contain the two variables [latex]xy[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"ohm146540\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146540&amp;theme=oea&amp;iframe_resize_id=ohm146540&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">Simplifying Expressions by Combining Like Terms<\/h2>\r\nWe can simplify an expression by combining the like terms. We saw this when adding fractions like [latex]\\frac{2}{\\color{blue}{7}}+\\frac{3}{\\color{blue}{7}}=\\frac{2+3}{\\color{blue}{7}}[\/latex]. In words this is [latex]2[\/latex]\u00a0<span style=\"color: #0000ff;\">sevenths<span style=\"color: #000000;\">\u00a0[latex]+3[\/latex]\u00a0<span style=\"color: #0000ff;\">sevenths<span style=\"color: #000000;\"> equals [latex]5[\/latex]\u00a0<span style=\"color: #0000ff;\">sevenths<span style=\"color: #000000;\">. Another example is\u00a0[latex]12[\/latex]\u00a0<span style=\"color: #ff6600;\">oranges<span style=\"color: #000000;\">\u00a0[latex]-8[\/latex]\u00a0<span style=\"color: #ff6600;\">oranges<span style=\"color: #000000;\">\u00a0[latex]=4[\/latex]\u00a0<span style=\"color: #ff6600;\">oranges<span style=\"color: #000000;\">.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nIf we apply the same reasoning to\u00a0[latex]3\\color{red}{x}+6\\color{red}{x}[\/latex] we add the coefficients\u00a0[latex]3+6[\/latex] and keep the like term\u00a0[latex]\\color{red}{x}[\/latex]:\u00a0\u00a0[latex]3\\color{red}{x}+6\\color{red}{x}=9\\color{red}{x}[\/latex]\r\n\r\nWe can see why this works by writing both terms as addition problems.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215839\/CNX_BMath_Figure_02_02_001_img.png\" alt=\"The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x.\" data-media-type=\"image\/png\" \/>\r\n\r\nAdd the coefficients and keep the common variable. It doesn\u2019t matter what [latex]x[\/latex] is. If we have [latex]3[\/latex] of something and add [latex]6[\/latex] more of the same thing, the result is [latex]9[\/latex] of them.\r\n\r\nThe expression [latex]3x+6x[\/latex] has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The <em><strong>Commutative Property of Addition<\/strong><\/em> says that we can change the order of addends without changing the sum, while the <em><strong>Associative Property of Addition\u00a0<\/strong><\/em>says we can regroup the addends without changing the sum. So we could rearrange and group the following expression before combining like terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215840\/CNX_BMath_Figure_02_02_015_img.png\" alt=\"The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together.\" data-media-type=\"image\/png\" \/>\r\n\r\nNow it is easier to see the like terms to be combined.\r\n<div class=\"textbox shaded\">\r\n<h3>Combining like terms<\/h3>\r\n<ol id=\"eip-id1168466010921\" class=\"stepwise\" data-number-style=\"arabic\">\r\n \t<li>Identify like terms.<\/li>\r\n \t<li>Rearrange the expression so like terms are together.<\/li>\r\n \t<li>Add the coefficients of the like terms and keep the common variable(s).<\/li>\r\n<\/ol>\r\n<\/div>\r\nIf we have like terms, we are\u00a0allowed to add (or subtract) the\u00a0numbers in front of the variables, then keep the variables the same. Kind of like saying four pens plus three pens equals seven pens. But two pens plus six pencils can\u2019t be combined and simplified because they are not \u201clike terms\u201d. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term. The sign always stays with the term.\r\n\r\nThis is shown in the following examples:\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify the expression: [latex]3x+7+4x+5[\/latex].\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n<h4>Solution:<\/h4>\r\n<table id=\"eip-id1168468738000\" class=\"unnumbered unstyled\" summary=\"The image shows the expression 3 x plus 7 plus 4 x plus 5. Three x and 4 x are like terms as are 7 and 5. The middle terms, 7 and 4 x, can be rearranged so that the like terms are together. The expressions becomes 3 x plus 4 x plus 7 plus 5. Now the like terms can be combined by adding the coefficients of the like terms. Three x plus 4 x is 7 x and 7 plus 5 is 12. The expression becomes 7 x plus 12.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3x+7+4x+5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Identify the like terms.<\/td>\r\n<td>[latex]\\color{red}{3x}+\\color{blue}{7}+\\color{red}{4x}+\\color{blue}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange the expression, so the like terms are together.<\/td>\r\n<td>[latex]\\color{red}{3x}+\\color{red}{4x}+\\color{blue}{7}+\\color{blue}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the coefficients of the like terms.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215843\/CNX_BMath_Figure_02_02_022_img-04.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The original expression is simplified to\u2026<\/td>\r\n<td>[latex]7x+12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nCombine like terms: \u00a0[latex]5x-2y-8x+7y[\/latex]\r\n<h4>Solution<\/h4>\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\nThe like terms in this expression are:\r\n<p style=\"text-align: center;\">[latex]5x[\/latex] and [latex]-8x[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]-2y[\/latex] and [latex]7y[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Note how we kept the sign in front of each term.<\/p>\r\n<p style=\"text-align: left;\">Combine like terms:<\/p>\r\n<p style=\"text-align: center;\">[latex]5x-8x = -3x[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]-2y+7y = 5y[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Note how signs become operations when we combine like terms.<\/p>\r\n<p style=\"text-align: left;\">Simplified Expression:<\/p>\r\n<p style=\"text-align: center;\">[latex]5x-2y-8x+7y=-3x+5y[\/latex]<\/p>\r\n<p style=\"text-align: center;\"><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"ohm144900\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144900&amp;theme=oea&amp;iframe_resize_id=ohm144900&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nCombine like terms: \u00a0[latex]x^2-3x+9-5x^2+3x-1[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n<h4>Solution<\/h4>\r\nThe like terms in this expression are:\r\n<p style=\"text-align: center;\">[latex]x^2[\/latex] and [latex]-5x^2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]-3x[\/latex] and [latex]3x[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]9[\/latex] and [latex]-1[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Combine like terms:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x^2-5x^2 = -4x^2\\\\-3x+3x=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\9-1=8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nSimplified Expression:\r\n<p style=\"text-align: center;\">[latex]-4x^2+8[\/latex]<\/p>\r\n<p style=\"text-align: center;\"><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe video that follows shows another example of\u00a0combining like terms. \u00a0Pay attention to why we are not able to combine all three terms in the example.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/b9-7eu29pNM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify the expression: [latex]8x+7{x}^{2}+{x}^{2}+4x[\/latex].\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n<h4>Solution:<\/h4>\r\n<table id=\"eip-id1168467195542\" class=\"unnumbered unstyled\" summary=\"The image shows the expression 7 x squared plus 8 x plus x squared plus 4 x. Seven x squared and x squared are like terms as are 8 x and 4 x. The middle terms, 8 x and x squared, can be rearranged so that the like terms are together. The expressions becomes 7 x squared plus x squared plus 8 x plus 4 x. Now the like terms can be combined by adding the coefficients of the like terms. 7 x squared plus x squared is 8 x squared and 8 x plus 4 x is 12 x. The expression becomes 8 x squared plus 12 x.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]8x+7{x}^{2}+{x}^{2}+4x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Identify the like terms.<\/td>\r\n<td>\u00a0[latex]\\color{blue}{8x}+\\color{red}{7x^2}+\\color{red}{x^2}+\\color{blue}{4x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange the expression so like terms are together.<\/td>\r\n<td>\u00a0[latex]\\color{red}{7x^2}+\\color{red}{x^2}+\\color{blue}{8x}+\\color{blue}{4x}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the coefficients of the like terms.<\/td>\r\n<td>\u00a0[latex]8x^2+12x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThese are not like terms and cannot be combined. So [latex]8{x}^{2}+12x[\/latex] is in simplest form.\r\n\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"ohm144905\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144905&amp;theme=oea&amp;iframe_resize_id=ohm144905&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\nThe following video presents more examples of how to combine like terms in an algebraic expression.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/KMUCQ_Pwt7o?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h2>Multiplication by a Constant and the Distributive Property<\/h2>\r\nMultiplying a term by a constant is the multiplication of the constant and the other elements of the term. For example, [latex]2\\cdot 3xy=2\\cdot 3\\cdot x\\cdot y[\/latex], which simplifies to [latex]6xy[\/latex] through multiplication of the constants. Likewise, [latex]-5\\cdot (-7x^2)=(-5\\cdot(-7))x^2=35x^2[\/latex] and [latex]-\\frac{2}{3}\\cdot 6x^2y^3=(-\\frac{2}{3}\\cdot 6)x^2y^3=-4x^2y^3[\/latex]. Here we use the<em><strong> Associative Property of Multiplication<\/strong><\/em> to regroup the multiplication of the constants.\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nSimplify:\r\n\r\n1. [latex]5\\cdot 4x[\/latex]\r\n\r\n2.\u00a0[latex]-2\\cdot 7x^2[\/latex]\r\n\r\n3. [latex]\\frac{2}{3}\\cdot 5xy[\/latex]\r\n\r\n4.\u00a0[latex]-\\frac{5}{6}\\cdot \\frac{4}{5}xy[\/latex]\r\n\r\n&nbsp;\r\n<h4>Solution<\/h4>\r\n1. [latex]5\\cdot 4x=(5\\cdot 4)x=20x[\/latex]\r\n\r\n2.\u00a0[latex]-2\\cdot 7x^2=(-2\\cdot 7)x^2=-14x^2[\/latex]\r\n\r\n3.\u00a0[latex]\\frac{2}{3}\\cdot 5xy=\\left (\\frac{2}{3}\\cdot 5\\right )xy=\\frac{10}{3}xy[\/latex]\r\n\r\n4.\u00a0[latex]-\\frac{5}{6}\\cdot \\frac{4}{5}xy=\\left (-\\frac{5}{6}\\cdot \\frac{4}{5}\\right )xy=-\\frac{5\\cdot 4}{6\\cdot 5}xy=-\\frac{5\\cdot 2\\cdot 2}{2\\cdot 3\\cdot 5}xy=-\\frac{2}{3}xy[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSimplify:\r\n\r\n1. [latex]8\\cdot 3x[\/latex]\r\n\r\n2.\u00a0[latex]2\\cdot -9x^3[\/latex]\r\n\r\n3. [latex]\\frac{4}{5}\\cdot 3xy[\/latex]\r\n\r\n4.\u00a0[latex]\\frac{4}{7}\\cdot \\frac{14}{5}x^4[\/latex]\r\n\r\n[reveal-answer q=\"430357\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"430357\"]\r\n<ol>\r\n \t<li>[latex]24x[\/latex]<\/li>\r\n \t<li>[latex]-18x^3[\/latex]<\/li>\r\n \t<li>[latex]\\frac{12}{5}xy[\/latex]<\/li>\r\n \t<li>[latex]\\frac{8}{5}x^4[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nTo multiply an expression that has two or more terms by a constant we have to multiply each term by that constant. This rule is called the <em><strong>Distributive Property of Multiplication over Addition<\/strong><\/em>. For example, to multiply the the expression [latex]3xy + 5z[\/latex] by [latex]2[\/latex] we must <em>distribute\u00a0<\/em>the\u00a0[latex]2[\/latex] to both [latex]3xy \\text{ and } 5z[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;= \\color{blue}{2}(3xy + 5z)\\\\ &amp;=\\color{blue}{2}\\cdot 3xy +\\color{blue}{2}\\cdot 5z\\\\ &amp;=6xy + 10z \\end{align}[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>The distributive property of multiplication over addition<\/h3>\r\n<p style=\"text-align: center;\">For any real number [latex]a[\/latex], [latex]a(x + y)=ax+ay[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nMultiply:\r\n\r\n1. [latex]5(4x+7y)[\/latex]\r\n\r\n2. [latex]-7(5x-y^2)[\/latex]\r\n\r\n3. [latex]\\frac{1}{2}(4x+6xy)[\/latex]\r\n\r\n4. [latex]-\\frac{4}{3}(6x^2+9x-2)[\/latex]\r\n\r\n5. [latex]-(5x-7)[\/latex]\r\n<h4>Solution<\/h4>\r\n1. [latex]\\color{blue}{5}(4x+7y)=\\color{blue}{5}\\cdot 4x +\\color{blue}{5}\\cdot 7y=20x + 35y[\/latex]\r\n\r\n2. [latex]\\color{blue}{-7}(5x-y^2)=\\color{blue}{-7}\\cdot 5x\\color{blue}{-7}\\cdot (-y^2)=-35x+7y^2[\/latex]\r\n\r\n3. [latex]\\color{blue}{\\frac{1}{2}}(4x-6xy)=\\color{blue}{\\frac{1}{2}}(4x)-\\color{blue}{\\frac{1}{2}}(6xy)=2x-3xy[\/latex]\r\n\r\n4. [latex]\\color{blue}{-\\frac{4}{3}}(6x^2+9x-2)=\\color{blue}{-\\frac{4}{3}}(6x^2)\\color{blue}{-\\frac{4}{3}}(9x)\\color{blue}{-\\frac{4}{3}}(-2)=-8x^2-12x+\\frac{8}{3}[\/latex]\r\n\r\n5.\u00a0[latex]\\color{blue}{-}(5x-7) =\\color{blue}{-1}(5x-7) =\\color{blue}{(-1)}5x - \\color{blue}{(-1)}7=-5x+7[\/latex]\r\n\r\n<\/div>\r\nExample 3 shows that fractions can be multiplied onto an expression. Multiplying by\u00a0[latex]\\frac{1}{2}[\/latex] is equivalent to dividing by [latex]2[\/latex]. This implies that the distributive property also applies to division:\r\n<p style=\"text-align: center;\">[latex]\\begin{align} &amp;= \\frac{6x+3}{\\color{blue}{3}}\\\\&amp;= \\frac{6x}{\\color{blue}{3}}+\\frac{3}{\\color{blue}{3}}\\\\ &amp;= 2x + 1\\end{align}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nDivide:\r\n\r\n1. [latex]\\frac{5x+30}{5}[\/latex]\r\n\r\n2.\u00a0[latex]\\frac{11x-44}{11}[\/latex]\r\n\r\n3.\u00a0[latex]\\frac{8x-40}{-8}[\/latex]\r\n\r\n4.\u00a0[latex]\\frac{-x+4}{-1}[\/latex]\r\n<h4>Solution<\/h4>\r\n1. [latex]\\frac{5x+30}{\\color{blue}{5}}=\\frac{5x}{\\color{blue}{5}}+\\frac{30}{\\color{blue}{5}}=x+6[\/latex]\r\n\r\n2.\u00a0[latex]\\frac{11x-44}{\\color{blue}{11}}=\\frac{11x}{\\color{blue}{11}}-\\frac{44}{\\color{blue}{11}}=x-4[\/latex]\r\n\r\n3.\u00a0[latex]\\frac{8x-40}{\\color{blue}{-8}}=\\frac{8x}{\\color{blue}{-8}}-\\frac{40}{\\color{blue}{-8}}=-x+5[\/latex]\r\n\r\n4.\u00a0[latex]\\frac{-x+4}{\\color{blue}{(-1)}}=\\frac{-x}{\\color{blue}{(-1)}}+\\frac{4}{\\color{blue}{(-1)}}=x-4[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nDivide:\r\n\r\n1. [latex]2(2x+40)[\/latex]\r\n\r\n2.\u00a0[latex]\\frac{15x-45}{5}[\/latex]\r\n\r\n3.\u00a0[latex]\\frac{12x-40}{-4}[\/latex]\r\n\r\n4. [latex]\\frac{2x-9}{-1}[\/latex]\r\n\r\n5. [latex]-3(5x-7)[\/latex]\r\n\r\n[reveal-answer q=\"hjm844\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm844\"]\r\n\r\n1. [latex]4x+80[\/latex]\r\n\r\n2.\u00a0[latex]3x-15[\/latex]\r\n\r\n3.\u00a0[latex]-3x+10[\/latex]\r\n\r\n4. [latex]-2x+9[\/latex]\r\n\r\n5. [latex]-15x+21[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<!-- edit page button -->\r\n\r\n<!-- page nav buttons -->","rendered":"<p><!-- Le HTML5 shim, for IE6-8 support of HTML5 elements --><!-- [if lt IE 9]>\n<script src=\"https:\/\/html5shim.googlecode.com\/svn\/trunk\/html5.js\">\n\t<\/script>\n<![endif] --><\/p>\n<div class=\"wrapper\">\n<div id=\"wrap\">\n<div id=\"content\" role=\"main\">\n<div id=\"post-67\" class=\"standard post-67 chapter type-chapter status-publish hentry\">\n<div class=\"entry-content\">\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the coefficient of a variable term<\/li>\n<li>Recognize and combine like terms in an expression<\/li>\n<li>Use the order of operations to simplify expressions containing like terms<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Term<\/strong>: a constant or the product or quotient of constants and one or more variables<\/li>\n<li><strong>Constant<\/strong>: a number that has a fixed value<\/li>\n<li><strong>Variable<\/strong>: a letter that represents a value that can change<\/li>\n<li><strong>Algebraic expression<\/strong>:\u00a0is a term, or the sum or difference of terms<\/li>\n<li><strong>Coefficient<\/strong>: the number in front of a variable or term<\/li>\n<li><strong>Like<\/strong> <strong>terms<\/strong>: terms that have the same variables and exponents<\/li>\n<li><strong>Commutative Property of Addition<\/strong>:\u00a0we can change the order of addends without changing the sum<\/li>\n<li><strong>Associative Property of Addition<\/strong>:\u00a0we can regroup the addends without changing the sum<\/li>\n<li><strong>Commutative Property of Multiplication<\/strong>: we can change the order of the factors without changing the product<\/li>\n<li><strong> Associative Property of Multiplication<\/strong>:\u00a0we can regroup the factors without changing the product<\/li>\n<li><strong>Distributive Property of Multiplication over Addition<\/strong>: a constant is multiplied onto an expression by multiplying each term in the expression by the constant<\/li>\n<\/ul>\n<\/div>\n<h2 data-type=\"title\">Identify Terms, Coefficients, and Like Terms<\/h2>\n<p>In mathematics, we may see expressions such as [latex]x+5[\/latex],\u00a0 [latex]\\dfrac{4}{3}{r}^{3}[\/latex], or [latex]5m-2n+6mn[\/latex]. Algebraic expressions are made up of\u00a0<strong>terms<\/strong>.<\/p>\n<p>A <em><strong>term<\/strong><\/em> is a constant or the product or quotient of constants and one or more variables. Some examples of terms are [latex]7,y,5{x}^{2},-\\frac{9a}{2b},\\text{ and }-13xy[\/latex].<\/p>\n<p>In the expression [latex]x+5[\/latex], [latex]5[\/latex] is called a\u00a0<strong>constant\u00a0<\/strong>because it does not vary and\u00a0[latex]x[\/latex]\u00a0is called a\u00a0<strong>variable\u00a0<\/strong>because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An\u00a0<strong><em>algebraic expression\u00a0<\/em><\/strong>is a term, or the sum of terms (remember that subtraction can always be written as a sum of the opposite).<\/p>\n<p>The constant that multiplies the variable(s) in a term is called the\u00a0<strong>coefficient<\/strong>. We can think of the coefficient as the number\u00a0<em data-effect=\"italics\">in front of\u00a0<\/em>the variable. The coefficient of the term [latex]3x[\/latex] is [latex]3[\/latex]. When we write [latex]x[\/latex], the coefficient is [latex]1[\/latex], since [latex]x=1\\cdot x[\/latex]. The table below gives the coefficients for each of the terms in the left column.<\/p>\n<table id=\"fs-id2266631\" summary=\"This table has five rows and two columns. The first row is a header row and it labels each column. The first column is labeled\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"center\">Term<\/th>\n<th data-align=\"center\">Coefficient<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]7[\/latex]<\/td>\n<td data-align=\"left\">[latex]7[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]-9a[\/latex]<\/td>\n<td data-align=\"left\">[latex]-9[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]y[\/latex]<\/td>\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]-\\frac{2}{5}{x}^{2}[\/latex]<\/td>\n<td data-align=\"left\">[latex]-\\frac{2}{5}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation with the term when we list it. Think of the operation as belonging to the term it precedes.<\/p>\n<table id=\"fs-id1596496\" summary=\"This table has six rows and two columns. The first row is a header row and it labels each column. The first column is labeled\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"center\">Expression<\/th>\n<th data-align=\"center\">Terms<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]-7[\/latex]<\/td>\n<td data-align=\"left\">[latex]-7[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]y[\/latex]<\/td>\n<td data-align=\"left\">[latex]y[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]x-\\frac{7}{5}[\/latex]<\/td>\n<td data-align=\"left\">[latex]x,-\\frac{7}{5}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]2x+7y-4[\/latex]<\/td>\n<td data-align=\"left\">[latex]2x,7y,-4[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]3{x}^{2}+4{x}^{2}+5y+3[\/latex]<\/td>\n<td data-align=\"left\">[latex]3{x}^{2},4{x}^{2},5y,3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div>Notice that when a term is being subtracted, the coefficient and the term are negative.<\/div>\n<div><\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Identify each term in the expression [latex]9b+15{x}^{2}-a+6[\/latex]. Then identify the coefficient of each term.<\/p>\n<p>Solution:<\/p>\n<p>The expression has four terms. They are [latex]9b,15{x}^{2},-a[\/latex], and [latex]6[\/latex].<\/p>\n<ul>\n<li>The coefficient of [latex]9b[\/latex] is [latex]9[\/latex].<\/li>\n<li>The coefficient of [latex]15{x}^{2}[\/latex] is [latex]15[\/latex].<\/li>\n<li>The coefficient of [latex]a[\/latex] is [latex]-1[\/latex].<\/li>\n<li>The coefficient of a constant is the constant, so the coefficient of [latex]6[\/latex] is [latex]6[\/latex].<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144899\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144899&amp;theme=oea&amp;iframe_resize_id=ohm144899&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>What exactly does it mean for a constant and a variable to be multiplied or divided? Let&#8217;s consider [latex]4x[\/latex]. Multiplication is repeated addition, so\u00a0[latex]4x=x+x+x+x[\/latex].\u00a0 What about [latex]\\frac{1}{5}x[\/latex]?\u00a0 Multiplying by\u00a0[latex]\\frac{1}{5}[\/latex] is equivalent to dividing by [latex]5[\/latex]. So,\u00a0[latex]\\frac{1}{5}x=\\frac{x}{5}[\/latex]. In other words, we take the quantity\u00a0[latex]x[\/latex] and divide it by[latex]5[\/latex].<\/p>\n<p>Recall that exponents are a more efficient way to write repeated multiplication. So, a variable with an exponent means to multiply the variable the exponent number of times. For example, [latex]x^3=x\\cdot x\\cdot x[\/latex] and [latex]y^5=y\\cdot y\\cdot y\\cdot y\\cdot y[\/latex].<\/p>\n<p>Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?<\/p>\n<p>[latex]5x,7,{n}^{2},4,-3x,9{n}^{2}[\/latex]<\/p>\n<ul id=\"fs-id1627987\" data-bullet-style=\"bullet\">\n<li>The terms [latex]7[\/latex] and [latex]4[\/latex] are both constant terms.<\/li>\n<li>The terms [latex]5x[\/latex] and [latex]-3x[\/latex] are both terms with [latex]x[\/latex].<\/li>\n<li>The terms [latex]{n}^{2}[\/latex] and [latex]9{n}^{2}[\/latex] both have [latex]{n}^{2}[\/latex].<\/li>\n<\/ul>\n<p>Terms are called <em><strong>like<\/strong> <strong>terms<\/strong><\/em> if they have the same variables and exponents. All constant terms are also like terms. So among the terms [latex]5x,7,{n}^{2},4,-3x,9{n}^{2}[\/latex],<\/p>\n<ul>\n<li>[latex]7[\/latex] and [latex]4[\/latex] are like terms.<\/li>\n<li>[latex]5x[\/latex] and [latex]3-x[\/latex] are like terms.<\/li>\n<li>[latex]{n}^{2}[\/latex] and [latex]9{n}^{2}[\/latex] are like terms.<\/li>\n<\/ul>\n<div class=\"textbox shaded\">\n<h3>Like Terms<\/h3>\n<p>Terms that are either constants or have the same variables with the same exponents are <strong>like terms<\/strong>.<\/p>\n<\/div>\n<p>Like terms\u00a0are terms where the variables match exactly (exponents included). Examples of like terms would be [latex]5xy[\/latex] and [latex]-3xy[\/latex], or [latex]8a^2b[\/latex] and [latex]a^2b[\/latex], or [latex]-\\frac{3}{4}[\/latex] and [latex]8[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Identify the like terms:<\/p>\n<ol>\n<li>[latex]{y}^{3},7{x}^{2},-14,\\frac{2}{3},4{y}^{3},-9x,5{x}^{2}[\/latex]<\/li>\n<li>[latex]-4{x}^{2}+2x+5{x}^{2}+6x-40x+8xy[\/latex]<\/li>\n<\/ol>\n<h4>Solution:<\/h4>\n<p>1. [latex]{y}^{3},7{x}^{2},-14,\\frac{2}{3},-4{y}^{3},-9x,5{x}^{2}[\/latex]<\/p>\n<p>Look at the variables and exponents. The expression contains [latex]{y}^{3},{x}^{2},x[\/latex], and constants.<\/p>\n<p>The terms [latex]{y}^{3}[\/latex] and [latex]-4{y}^{3}[\/latex] are like terms because they both contain [latex]{y}^{3}[\/latex].<\/p>\n<p>The terms [latex]7{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both contain [latex]{x}^{2}[\/latex].<\/p>\n<p>The terms [latex]-14[\/latex] and [latex]\\frac{2}{3}[\/latex] are like terms because they are both constants.<\/p>\n<p>The term [latex]-9x[\/latex] does not have any like terms in this list since no other terms have the variable [latex]x[\/latex] raised to the power of [latex]1[\/latex].<\/p>\n<p>2. [latex]-4{x}^{2}+2x+5{x}^{2}+6x-40x+8xy[\/latex]<\/p>\n<p>Look at the variables and exponents. The expression contains [latex]{x}^{2},x,text{and}xy[\/latex] terms.<\/p>\n<p>The terms [latex]-4{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both contain [latex]{x}^{2}[\/latex].<\/p>\n<p>The terms [latex]2x,6x,\\text{and}-40x[\/latex] are like terms because they all contain [latex]x[\/latex].<\/p>\n<p>The term [latex]8xy[\/latex] has no like terms in the given expression because no other terms contain the two variables [latex]xy[\/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=\"ohm146540\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146540&amp;theme=oea&amp;iframe_resize_id=ohm146540&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2 data-type=\"title\">Simplifying Expressions by Combining Like Terms<\/h2>\n<p>We can simplify an expression by combining the like terms. We saw this when adding fractions like [latex]\\frac{2}{\\color{blue}{7}}+\\frac{3}{\\color{blue}{7}}=\\frac{2+3}{\\color{blue}{7}}[\/latex]. In words this is [latex]2[\/latex]\u00a0<span style=\"color: #0000ff;\">sevenths<span style=\"color: #000000;\">\u00a0[latex]+3[\/latex]\u00a0<span style=\"color: #0000ff;\">sevenths<span style=\"color: #000000;\"> equals [latex]5[\/latex]\u00a0<span style=\"color: #0000ff;\">sevenths<span style=\"color: #000000;\">. Another example is\u00a0[latex]12[\/latex]\u00a0<span style=\"color: #ff6600;\">oranges<span style=\"color: #000000;\">\u00a0[latex]-8[\/latex]\u00a0<span style=\"color: #ff6600;\">oranges<span style=\"color: #000000;\">\u00a0[latex]=4[\/latex]\u00a0<span style=\"color: #ff6600;\">oranges<span style=\"color: #000000;\">.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>If we apply the same reasoning to\u00a0[latex]3\\color{red}{x}+6\\color{red}{x}[\/latex] we add the coefficients\u00a0[latex]3+6[\/latex] and keep the like term\u00a0[latex]\\color{red}{x}[\/latex]:\u00a0\u00a0[latex]3\\color{red}{x}+6\\color{red}{x}=9\\color{red}{x}[\/latex]<\/p>\n<p>We can see why this works by writing both terms as addition problems.<\/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\/24215839\/CNX_BMath_Figure_02_02_001_img.png\" alt=\"The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x.\" data-media-type=\"image\/png\" \/><\/p>\n<p>Add the coefficients and keep the common variable. It doesn\u2019t matter what [latex]x[\/latex] is. If we have [latex]3[\/latex] of something and add [latex]6[\/latex] more of the same thing, the result is [latex]9[\/latex] of them.<\/p>\n<p>The expression [latex]3x+6x[\/latex] has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The <em><strong>Commutative Property of Addition<\/strong><\/em> says that we can change the order of addends without changing the sum, while the <em><strong>Associative Property of Addition\u00a0<\/strong><\/em>says we can regroup the addends without changing the sum. So we could rearrange and group the following expression before combining like terms.<\/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\/24215840\/CNX_BMath_Figure_02_02_015_img.png\" alt=\"The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together.\" data-media-type=\"image\/png\" \/><\/p>\n<p>Now it is easier to see the like terms to be combined.<\/p>\n<div class=\"textbox shaded\">\n<h3>Combining like terms<\/h3>\n<ol id=\"eip-id1168466010921\" class=\"stepwise\" data-number-style=\"arabic\">\n<li>Identify like terms.<\/li>\n<li>Rearrange the expression so like terms are together.<\/li>\n<li>Add the coefficients of the like terms and keep the common variable(s).<\/li>\n<\/ol>\n<\/div>\n<p>If we have like terms, we are\u00a0allowed to add (or subtract) the\u00a0numbers in front of the variables, then keep the variables the same. Kind of like saying four pens plus three pens equals seven pens. But two pens plus six pencils can\u2019t be combined and simplified because they are not \u201clike terms\u201d. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term. The sign always stays with the term.<\/p>\n<p>This is shown in the following examples:<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify the expression: [latex]3x+7+4x+5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<h4>Solution:<\/h4>\n<table id=\"eip-id1168468738000\" class=\"unnumbered unstyled\" summary=\"The image shows the expression 3 x plus 7 plus 4 x plus 5. Three x and 4 x are like terms as are 7 and 5. The middle terms, 7 and 4 x, can be rearranged so that the like terms are together. The expressions becomes 3 x plus 4 x plus 7 plus 5. Now the like terms can be combined by adding the coefficients of the like terms. Three x plus 4 x is 7 x and 7 plus 5 is 12. The expression becomes 7 x plus 12.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3x+7+4x+5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Identify the like terms.<\/td>\n<td>[latex]\\color{red}{3x}+\\color{blue}{7}+\\color{red}{4x}+\\color{blue}{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rearrange the expression, so the like terms are together.<\/td>\n<td>[latex]\\color{red}{3x}+\\color{red}{4x}+\\color{blue}{7}+\\color{blue}{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the coefficients of the like terms.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215843\/CNX_BMath_Figure_02_02_022_img-04.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>The original expression is simplified to\u2026<\/td>\n<td>[latex]7x+12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Combine like terms: \u00a0[latex]5x-2y-8x+7y[\/latex]<\/p>\n<h4>Solution<\/h4>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p>The like terms in this expression are:<\/p>\n<p style=\"text-align: center;\">[latex]5x[\/latex] and [latex]-8x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]-2y[\/latex] and [latex]7y[\/latex]<\/p>\n<p style=\"text-align: left;\">Note how we kept the sign in front of each term.<\/p>\n<p style=\"text-align: left;\">Combine like terms:<\/p>\n<p style=\"text-align: center;\">[latex]5x-8x = -3x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]-2y+7y = 5y[\/latex]<\/p>\n<p style=\"text-align: left;\">Note how signs become operations when we combine like terms.<\/p>\n<p style=\"text-align: left;\">Simplified Expression:<\/p>\n<p style=\"text-align: center;\">[latex]5x-2y-8x+7y=-3x+5y[\/latex]<\/p>\n<p style=\"text-align: center;\">\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144900\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144900&amp;theme=oea&amp;iframe_resize_id=ohm144900&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Combine like terms: \u00a0[latex]x^2-3x+9-5x^2+3x-1[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<h4>Solution<\/h4>\n<p>The like terms in this expression are:<\/p>\n<p style=\"text-align: center;\">[latex]x^2[\/latex] and [latex]-5x^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]-3x[\/latex] and [latex]3x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]9[\/latex] and [latex]-1[\/latex]<\/p>\n<p style=\"text-align: left;\">Combine like terms:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x^2-5x^2 = -4x^2\\\\-3x+3x=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\9-1=8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Simplified Expression:<\/p>\n<p style=\"text-align: center;\">[latex]-4x^2+8[\/latex]<\/p>\n<p style=\"text-align: center;\">\n<\/div>\n<\/div>\n<\/div>\n<p>The video that follows shows another example of\u00a0combining like terms. \u00a0Pay attention to why we are not able to combine all three terms in the example.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/b9-7eu29pNM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify the expression: [latex]8x+7{x}^{2}+{x}^{2}+4x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<h4>Solution:<\/h4>\n<table id=\"eip-id1168467195542\" class=\"unnumbered unstyled\" summary=\"The image shows the expression 7 x squared plus 8 x plus x squared plus 4 x. Seven x squared and x squared are like terms as are 8 x and 4 x. The middle terms, 8 x and x squared, can be rearranged so that the like terms are together. The expressions becomes 7 x squared plus x squared plus 8 x plus 4 x. Now the like terms can be combined by adding the coefficients of the like terms. 7 x squared plus x squared is 8 x squared and 8 x plus 4 x is 12 x. The expression becomes 8 x squared plus 12 x.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]8x+7{x}^{2}+{x}^{2}+4x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Identify the like terms.<\/td>\n<td>\u00a0[latex]\\color{blue}{8x}+\\color{red}{7x^2}+\\color{red}{x^2}+\\color{blue}{4x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rearrange the expression so like terms are together.<\/td>\n<td>\u00a0[latex]\\color{red}{7x^2}+\\color{red}{x^2}+\\color{blue}{8x}+\\color{blue}{4x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the coefficients of the like terms.<\/td>\n<td>\u00a0[latex]8x^2+12x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These are not like terms and cannot be combined. So [latex]8{x}^{2}+12x[\/latex] is in simplest form.<\/p>\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=\"ohm144905\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144905&amp;theme=oea&amp;iframe_resize_id=ohm144905&amp;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video presents more examples of how to combine like terms in an algebraic expression.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/KMUCQ_Pwt7o?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiplication by a Constant and the Distributive Property<\/h2>\n<p>Multiplying a term by a constant is the multiplication of the constant and the other elements of the term. For example, [latex]2\\cdot 3xy=2\\cdot 3\\cdot x\\cdot y[\/latex], which simplifies to [latex]6xy[\/latex] through multiplication of the constants. Likewise, [latex]-5\\cdot (-7x^2)=(-5\\cdot(-7))x^2=35x^2[\/latex] and [latex]-\\frac{2}{3}\\cdot 6x^2y^3=(-\\frac{2}{3}\\cdot 6)x^2y^3=-4x^2y^3[\/latex]. Here we use the<em><strong> Associative Property of Multiplication<\/strong><\/em> to regroup the multiplication of the constants.<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]5\\cdot 4x[\/latex]<\/p>\n<p>2.\u00a0[latex]-2\\cdot 7x^2[\/latex]<\/p>\n<p>3. [latex]\\frac{2}{3}\\cdot 5xy[\/latex]<\/p>\n<p>4.\u00a0[latex]-\\frac{5}{6}\\cdot \\frac{4}{5}xy[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<h4>Solution<\/h4>\n<p>1. [latex]5\\cdot 4x=(5\\cdot 4)x=20x[\/latex]<\/p>\n<p>2.\u00a0[latex]-2\\cdot 7x^2=(-2\\cdot 7)x^2=-14x^2[\/latex]<\/p>\n<p>3.\u00a0[latex]\\frac{2}{3}\\cdot 5xy=\\left (\\frac{2}{3}\\cdot 5\\right )xy=\\frac{10}{3}xy[\/latex]<\/p>\n<p>4.\u00a0[latex]-\\frac{5}{6}\\cdot \\frac{4}{5}xy=\\left (-\\frac{5}{6}\\cdot \\frac{4}{5}\\right )xy=-\\frac{5\\cdot 4}{6\\cdot 5}xy=-\\frac{5\\cdot 2\\cdot 2}{2\\cdot 3\\cdot 5}xy=-\\frac{2}{3}xy[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Simplify:<\/p>\n<p>1. [latex]8\\cdot 3x[\/latex]<\/p>\n<p>2.\u00a0[latex]2\\cdot -9x^3[\/latex]<\/p>\n<p>3. [latex]\\frac{4}{5}\\cdot 3xy[\/latex]<\/p>\n<p>4.\u00a0[latex]\\frac{4}{7}\\cdot \\frac{14}{5}x^4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q430357\">Show Answer<\/span><\/p>\n<div id=\"q430357\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]24x[\/latex]<\/li>\n<li>[latex]-18x^3[\/latex]<\/li>\n<li>[latex]\\frac{12}{5}xy[\/latex]<\/li>\n<li>[latex]\\frac{8}{5}x^4[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>To multiply an expression that has two or more terms by a constant we have to multiply each term by that constant. This rule is called the <em><strong>Distributive Property of Multiplication over Addition<\/strong><\/em>. For example, to multiply the the expression [latex]3xy + 5z[\/latex] by [latex]2[\/latex] we must <em>distribute\u00a0<\/em>the\u00a0[latex]2[\/latex] to both [latex]3xy \\text{ and } 5z[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &= \\color{blue}{2}(3xy + 5z)\\\\ &=\\color{blue}{2}\\cdot 3xy +\\color{blue}{2}\\cdot 5z\\\\ &=6xy + 10z \\end{align}[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>The distributive property of multiplication over addition<\/h3>\n<p style=\"text-align: center;\">For any real number [latex]a[\/latex], [latex]a(x + y)=ax+ay[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Multiply:<\/p>\n<p>1. [latex]5(4x+7y)[\/latex]<\/p>\n<p>2. [latex]-7(5x-y^2)[\/latex]<\/p>\n<p>3. [latex]\\frac{1}{2}(4x+6xy)[\/latex]<\/p>\n<p>4. [latex]-\\frac{4}{3}(6x^2+9x-2)[\/latex]<\/p>\n<p>5. [latex]-(5x-7)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>1. [latex]\\color{blue}{5}(4x+7y)=\\color{blue}{5}\\cdot 4x +\\color{blue}{5}\\cdot 7y=20x + 35y[\/latex]<\/p>\n<p>2. [latex]\\color{blue}{-7}(5x-y^2)=\\color{blue}{-7}\\cdot 5x\\color{blue}{-7}\\cdot (-y^2)=-35x+7y^2[\/latex]<\/p>\n<p>3. [latex]\\color{blue}{\\frac{1}{2}}(4x-6xy)=\\color{blue}{\\frac{1}{2}}(4x)-\\color{blue}{\\frac{1}{2}}(6xy)=2x-3xy[\/latex]<\/p>\n<p>4. [latex]\\color{blue}{-\\frac{4}{3}}(6x^2+9x-2)=\\color{blue}{-\\frac{4}{3}}(6x^2)\\color{blue}{-\\frac{4}{3}}(9x)\\color{blue}{-\\frac{4}{3}}(-2)=-8x^2-12x+\\frac{8}{3}[\/latex]<\/p>\n<p>5.\u00a0[latex]\\color{blue}{-}(5x-7) =\\color{blue}{-1}(5x-7) =\\color{blue}{(-1)}5x - \\color{blue}{(-1)}7=-5x+7[\/latex]<\/p>\n<\/div>\n<p>Example 3 shows that fractions can be multiplied onto an expression. Multiplying by\u00a0[latex]\\frac{1}{2}[\/latex] is equivalent to dividing by [latex]2[\/latex]. This implies that the distributive property also applies to division:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} &= \\frac{6x+3}{\\color{blue}{3}}\\\\&= \\frac{6x}{\\color{blue}{3}}+\\frac{3}{\\color{blue}{3}}\\\\ &= 2x + 1\\end{align}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Divide:<\/p>\n<p>1. [latex]\\frac{5x+30}{5}[\/latex]<\/p>\n<p>2.\u00a0[latex]\\frac{11x-44}{11}[\/latex]<\/p>\n<p>3.\u00a0[latex]\\frac{8x-40}{-8}[\/latex]<\/p>\n<p>4.\u00a0[latex]\\frac{-x+4}{-1}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>1. [latex]\\frac{5x+30}{\\color{blue}{5}}=\\frac{5x}{\\color{blue}{5}}+\\frac{30}{\\color{blue}{5}}=x+6[\/latex]<\/p>\n<p>2.\u00a0[latex]\\frac{11x-44}{\\color{blue}{11}}=\\frac{11x}{\\color{blue}{11}}-\\frac{44}{\\color{blue}{11}}=x-4[\/latex]<\/p>\n<p>3.\u00a0[latex]\\frac{8x-40}{\\color{blue}{-8}}=\\frac{8x}{\\color{blue}{-8}}-\\frac{40}{\\color{blue}{-8}}=-x+5[\/latex]<\/p>\n<p>4.\u00a0[latex]\\frac{-x+4}{\\color{blue}{(-1)}}=\\frac{-x}{\\color{blue}{(-1)}}+\\frac{4}{\\color{blue}{(-1)}}=x-4[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Divide:<\/p>\n<p>1. [latex]2(2x+40)[\/latex]<\/p>\n<p>2.\u00a0[latex]\\frac{15x-45}{5}[\/latex]<\/p>\n<p>3.\u00a0[latex]\\frac{12x-40}{-4}[\/latex]<\/p>\n<p>4. [latex]\\frac{2x-9}{-1}[\/latex]<\/p>\n<p>5. [latex]-3(5x-7)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm844\">Show Answer<\/span><\/p>\n<div id=\"qhjm844\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]4x+80[\/latex]<\/p>\n<p>2.\u00a0[latex]3x-15[\/latex]<\/p>\n<p>3.\u00a0[latex]-3x+10[\/latex]<\/p>\n<p>4. [latex]-2x+9[\/latex]<\/p>\n<p>5. [latex]-15x+21[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p><!-- edit page button --><\/p>\n<p><!-- page nav buttons --><\/p>\n","protected":false},"author":370291,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2830","chapter","type-chapter","status-publish","hentry"],"part":2825,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2830","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/370291"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2830\/revisions"}],"predecessor-version":[{"id":2831,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2830\/revisions\/2831"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/2825"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/2830\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=2830"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2830"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=2830"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=2830"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}