{"id":9241,"date":"2017-05-02T19:51:09","date_gmt":"2017-05-02T19:51:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9241"},"modified":"2017-09-23T15:34:30","modified_gmt":"2017-09-23T15:34:30","slug":"simplifying-algebraic-expressions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/simplifying-algebraic-expressions\/","title":{"raw":"Simplifying Algebraic Expressions","rendered":"Simplifying Algebraic Expressions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the variables and constants in a term<\/li>\r\n \t<li>Identify the coefficient of a variable term<\/li>\r\n \t<li>Identify and combine like terms in an expression<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 data-type=\"title\">Identify Terms, Coefficients, and Like Terms<\/h2>\r\nAlgebraic expressions are made up of <em data-effect=\"italics\">terms<\/em>. A term is a constant or the product of a constant and one or more variables. Some examples of terms are [latex]7,y,5{x}^{2},9a,\\text{and }13xy[\/latex].\r\n\r\nThe constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number <em data-effect=\"italics\">in front of<\/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]5{x}^{2}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAn algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. The table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.\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+7[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]x,7[\/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><\/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\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>Remember that if no number is written before a variable, the coefficient is [latex]1[\/latex]. So 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&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]144899[\/ohm_question]\r\n\r\n<\/div>\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\nWhich of these terms are like terms?\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 like terms 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]3x[\/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 like terms.\r\n\r\n<\/div>\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},14,23,4{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[reveal-answer q=\"169480\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"169480\"]\r\n\r\nSolution:\r\n1. [latex]{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}[\/latex]\r\nLook at the variables and exponents. The expression contains [latex]{y}^{3},{x}^{2},x[\/latex], and constants.\r\nThe terms [latex]{y}^{3}[\/latex] and [latex]4{y}^{3}[\/latex] are like terms because they both have [latex]{y}^{3}[\/latex].\r\nThe terms [latex]7{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both have [latex]{x}^{2}[\/latex].\r\nThe terms [latex]14[\/latex] and [latex]23[\/latex] are like terms because they are both constants.\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\nLook at the variables and exponents. The expression contains the terms [latex]4{x}^{2},2x,5{x}^{2},6x,40x,\\text{and}8xy[\/latex]\r\nThe terms [latex]4{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both have [latex]{x}^{2}[\/latex].\r\nThe terms [latex]2x,6x,\\text{and}40x[\/latex] are like terms because they all have [latex]x[\/latex].\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[\/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]146540[\/ohm_question]\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">Simplify Expressions by Combining Like Terms<\/h2>\r\nWe can simplify an expression by combining the like terms. What do you think [latex]3x+6x[\/latex] would simplify to? If you thought [latex]9x[\/latex], you would be right!\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\nAdd the coefficients and keep the same variable. It doesn\u2019t matter what [latex]x[\/latex] is. If you have [latex]3[\/latex] of something and add [latex]6[\/latex] more of the same thing, the result is [latex]9[\/latex] of them. For example, [latex]3[\/latex] oranges plus [latex]6[\/latex] oranges is [latex]9[\/latex] oranges. We will discuss the mathematical properties behind this later.\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 Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange 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\nNow it is easier to see the like terms to be combined.\r\n<div class=\"textbox shaded\">\r\n<h3>Combine 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.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify the expression: [latex]3x+7+4x+5[\/latex].\r\n[reveal-answer q=\"668173\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"668173\"]\r\n\r\nSolution:\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...<\/td>\r\n<td>[latex]7x+12[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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]144900[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\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[reveal-answer q=\"94190\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"94190\"]\r\n\r\nSolution:\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[\/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]144905[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video, we present more examples of how to combine like terms given an algebraic expression.\r\n\r\nhttps:\/\/youtu.be\/KMUCQ_Pwt7o","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the variables and constants in a term<\/li>\n<li>Identify the coefficient of a variable term<\/li>\n<li>Identify and combine like terms in an expression<\/li>\n<\/ul>\n<\/div>\n<h2 data-type=\"title\">Identify Terms, Coefficients, and Like Terms<\/h2>\n<p>Algebraic expressions are made up of <em data-effect=\"italics\">terms<\/em>. A term is a constant or the product of a constant and one or more variables. Some examples of terms are [latex]7,y,5{x}^{2},9a,\\text{and }13xy[\/latex].<\/p>\n<p>The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number <em data-effect=\"italics\">in front of<\/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]5{x}^{2}[\/latex]<\/td>\n<td data-align=\"left\">[latex]5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. The table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.<\/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+7[\/latex]<\/td>\n<td data-align=\"left\">[latex]x,7[\/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><\/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:<br \/>\nThe 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>Remember that if no number is written before a variable, the coefficient is [latex]1[\/latex]. So 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<p>&nbsp;<\/p>\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&theme=oea&iframe_resize_id=ohm144899&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\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]<br \/>\nWhich of these terms are like terms?<\/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 like terms 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]3x[\/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 like terms.<\/p>\n<\/div>\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},14,23,4{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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q169480\">Show Solution<\/span><\/p>\n<div id=\"q169480\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\n1. [latex]{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}[\/latex]<br \/>\nLook at the variables and exponents. The expression contains [latex]{y}^{3},{x}^{2},x[\/latex], and constants.<br \/>\nThe terms [latex]{y}^{3}[\/latex] and [latex]4{y}^{3}[\/latex] are like terms because they both have [latex]{y}^{3}[\/latex].<br \/>\nThe terms [latex]7{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both have [latex]{x}^{2}[\/latex].<br \/>\nThe terms [latex]14[\/latex] and [latex]23[\/latex] are like terms because they are both constants.<br \/>\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].<\/p>\n<p>2. [latex]4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy[\/latex]<br \/>\nLook at the variables and exponents. The expression contains the terms [latex]4{x}^{2},2x,5{x}^{2},6x,40x,\\text{and}8xy[\/latex]<br \/>\nThe terms [latex]4{x}^{2}[\/latex] and [latex]5{x}^{2}[\/latex] are like terms because they both have [latex]{x}^{2}[\/latex].<br \/>\nThe terms [latex]2x,6x,\\text{and}40x[\/latex] are like terms because they all have [latex]x[\/latex].<br \/>\nThe 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&theme=oea&iframe_resize_id=ohm146540&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2 data-type=\"title\">Simplify Expressions by Combining Like Terms<\/h2>\n<p>We can simplify an expression by combining the like terms. What do you think [latex]3x+6x[\/latex] would simplify to? If you thought [latex]9x[\/latex], you would be right!<\/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\" \/><br \/>\nAdd the coefficients and keep the same variable. It doesn\u2019t matter what [latex]x[\/latex] is. If you have [latex]3[\/latex] of something and add [latex]6[\/latex] more of the same thing, the result is [latex]9[\/latex] of them. For example, [latex]3[\/latex] oranges plus [latex]6[\/latex] oranges is [latex]9[\/latex] oranges. We will discuss the mathematical properties behind this later.<\/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 Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange 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\" \/><br \/>\nNow it is easier to see the like terms to be combined.<\/p>\n<div class=\"textbox shaded\">\n<h3>Combine 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.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q668173\">Show Solution<\/span><\/p>\n<div id=\"q668173\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\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&#8230;<\/td>\n<td>[latex]7x+12[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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=\"ohm144900\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144900&theme=oea&iframe_resize_id=ohm144900&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>Simplify the expression: [latex]8x+7{x}^{2}+{x}^{2}+4x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q94190\">Show Solution<\/span><\/p>\n<div id=\"q94190\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\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<\/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&theme=oea&iframe_resize_id=ohm144905&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video, we present more examples of how to combine like terms given an algebraic expression.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify Expressions by Combining Like Terms (No Negatives)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KMUCQ_Pwt7o?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-9241\">\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>Simplify Expressions by Combining Like Terms (No Negatives). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KMUCQ_Pwt7o\">https:\/\/youtu.be\/KMUCQ_Pwt7o<\/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, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Combining Like Terms. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/JIleqbO8Tf0\">https:\/\/youtu.be\/JIleqbO8Tf0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 2: Combining Like Terms. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/b9-7eu29pNM\">https:\/\/youtu.be\/b9-7eu29pNM<\/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>Question ID: 144899, 144900, 144905,146540. <strong>Authored by<\/strong>: Alyson Day. <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, 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":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Combining Like Terms\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/JIleqbO8Tf0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 2: Combining Like Terms\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/b9-7eu29pNM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"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\":\"original\",\"description\":\"Simplify Expressions by Combining Like Terms (No Negatives)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/KMUCQ_Pwt7o\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID: 144899, 144900, 144905,146540\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + 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