{"id":9343,"date":"2017-05-02T21:49:13","date_gmt":"2017-05-02T21:49:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9343"},"modified":"2017-09-23T20:15:48","modified_gmt":"2017-09-23T20:15:48","slug":"determining-whether-an-integer-is-a-solution-to-an-equation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-prealgebra\/chapter\/determining-whether-an-integer-is-a-solution-to-an-equation\/","title":{"raw":"Determining Whether an Integer is a Solution to an Equation","rendered":"Determining Whether an Integer is a Solution to an Equation"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine whether an integer is a solution to an equation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Determine Whether an Integer is a Solution of an Equation<\/h2>\r\nIn Solve Equations with the Subtraction and Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we\u2019ve worked with integers, we\u2019ll find integer solutions to equations.\r\n\r\nThe steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.\r\n<div class=\"textbox shaded\">\r\n<h3>How to determine whether a number is a solution to an equation<\/h3>\r\n<ol id=\"eip-id1168469806892\" class=\"stepwise\">\r\n \t<li>Substitute the number for the variable in the equation.<\/li>\r\n \t<li>Simplify the expressions on both sides of the equation.<\/li>\r\n \t<li>Determine whether the resulting equation is true.\r\n<ul id=\"fs-id1787822\">\r\n \t<li>If it is true, the number is a solution.<\/li>\r\n \t<li>If it is not true, the number is not a solution.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDetermine whether each of the following is a solution of [latex]2x - 5=-13\\text{:}[\/latex]\r\n\r\n1. [latex]x=4[\/latex]\r\n2. [latex]x=-4[\/latex]\r\n3. [latex]x=-9[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168468249954\" class=\"unnumbered unstyled\" summary=\"This figure has 4 rows. The first row is 2 times x minus 5 equals negative 13. The second row states substitute 4 for x and has 2 times 4 minus 5 equals negative 13. The third row states multiply and has 8 minus 5 equals negative 13. The fourth row states subtract and has 3 is not equal to negative 13.\">\r\n<tbody>\r\n<tr>\r\n<td>1. Substitute [latex]4[\/latex] for x in the equation to determine if it is true.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]2x--5=--13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0Substitute [latex]\\color{red}{4}[\/latex] for x.<\/td>\r\n<td>[latex]2(\\color{red}{4})--5=--13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]8--5=--13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]3\\not=--13[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]x=4[\/latex] does not result in a true equation, [latex]4[\/latex] is not a solution to the equation.\r\n<table id=\"eip-id1168469614488\" class=\"unnumbered unstyled\" summary=\"This figure has 4 rows. The first row is 2 times x minus 5 equals negative 13. The second row states substitute negative 4 for x and has 2 times negative 4 minus 5 equals negative 13. The third row states multiply and has negative 8 minus 5 equals negative 13. The fourth row states subtract and has negative 13 equals negative 13.\">\r\n<tbody>\r\n<tr>\r\n<td>2. Substitute [latex]\u22124[\/latex] for x in the equation to determine if it is true.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]2x--5=--13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{--4}[\/latex] for x.<\/td>\r\n<td>[latex]2(\\color{red}{-4})--5=--13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]--8--5=--13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]--13=--13\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]x=-4[\/latex] results in a true equation, [latex]-4[\/latex] is a solution to the equation.\r\n<table id=\"eip-id1168468223504\" class=\"unnumbered unstyled\" summary=\"This figure has 4 rows. The first row is 2 times x minus 5 equals negative 13. The second row states substitute negative 9 for x and has 2 times negative 9 minus 5 equals negative 13 with a question mark above the equal sign. The third row states multiply and has negative 18 minus 5 equals negative 13 with a question mark above the equal sign. The fourth row states subtract and has negative 23 is not equal to negative 13.\">\r\n<tbody>\r\n<tr>\r\n<td>3. Substitute [latex]\u22129[\/latex] for x in the equation to determine if it is true.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]2x--5=--13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\u22129[\/latex] for x.<\/td>\r\n<td>[latex]2(\\color{red}{--9})--5=--13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]--18--5=--13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]--23\\not=--13[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]x=-9[\/latex] does not result in a true equation, [latex]-9[\/latex] is not a solution to the equation.\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146556[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we present more examples of how to verify that an integer is a solution to a linear equation.\r\nhttps:\/\/youtu.be\/eBameNAndKw","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine whether an integer is a solution to an equation<\/li>\n<\/ul>\n<\/div>\n<h2>Determine Whether an Integer is a Solution of an Equation<\/h2>\n<p>In Solve Equations with the Subtraction and Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we\u2019ve worked with integers, we\u2019ll find integer solutions to equations.<\/p>\n<p>The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.<\/p>\n<div class=\"textbox shaded\">\n<h3>How to determine whether a number is a solution to an equation<\/h3>\n<ol id=\"eip-id1168469806892\" class=\"stepwise\">\n<li>Substitute the number for the variable in the equation.<\/li>\n<li>Simplify the expressions on both sides of the equation.<\/li>\n<li>Determine whether the resulting equation is true.\n<ul id=\"fs-id1787822\">\n<li>If it is true, the number is a solution.<\/li>\n<li>If it is not true, the number is not a solution.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Determine whether each of the following is a solution of [latex]2x - 5=-13\\text{:}[\/latex]<\/p>\n<p>1. [latex]x=4[\/latex]<br \/>\n2. [latex]x=-4[\/latex]<br \/>\n3. [latex]x=-9[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468249954\" class=\"unnumbered unstyled\" summary=\"This figure has 4 rows. The first row is 2 times x minus 5 equals negative 13. The second row states substitute 4 for x and has 2 times 4 minus 5 equals negative 13. The third row states multiply and has 8 minus 5 equals negative 13. The fourth row states subtract and has 3 is not equal to negative 13.\">\n<tbody>\n<tr>\n<td>1. Substitute [latex]4[\/latex] for x in the equation to determine if it is true.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]2x--5=--13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0Substitute [latex]\\color{red}{4}[\/latex] for x.<\/td>\n<td>[latex]2(\\color{red}{4})--5=--13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]8--5=--13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]3\\not=--13[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]x=4[\/latex] does not result in a true equation, [latex]4[\/latex] is not a solution to the equation.<\/p>\n<table id=\"eip-id1168469614488\" class=\"unnumbered unstyled\" summary=\"This figure has 4 rows. The first row is 2 times x minus 5 equals negative 13. The second row states substitute negative 4 for x and has 2 times negative 4 minus 5 equals negative 13. The third row states multiply and has negative 8 minus 5 equals negative 13. The fourth row states subtract and has negative 13 equals negative 13.\">\n<tbody>\n<tr>\n<td>2. Substitute [latex]\u22124[\/latex] for x in the equation to determine if it is true.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]2x--5=--13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{--4}[\/latex] for x.<\/td>\n<td>[latex]2(\\color{red}{-4})--5=--13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]--8--5=--13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]--13=--13\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]x=-4[\/latex] results in a true equation, [latex]-4[\/latex] is a solution to the equation.<\/p>\n<table id=\"eip-id1168468223504\" class=\"unnumbered unstyled\" summary=\"This figure has 4 rows. The first row is 2 times x minus 5 equals negative 13. The second row states substitute negative 9 for x and has 2 times negative 9 minus 5 equals negative 13 with a question mark above the equal sign. The third row states multiply and has negative 18 minus 5 equals negative 13 with a question mark above the equal sign. The fourth row states subtract and has negative 23 is not equal to negative 13.\">\n<tbody>\n<tr>\n<td>3. Substitute [latex]\u22129[\/latex] for x in the equation to determine if it is true.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]2x--5=--13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\u22129[\/latex] for x.<\/td>\n<td>[latex]2(\\color{red}{--9})--5=--13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]--18--5=--13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]--23\\not=--13[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]x=-9[\/latex] does not result in a true equation, [latex]-9[\/latex] is not a solution to the equation.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146556\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146556&theme=oea&iframe_resize_id=ohm146556&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we present more examples of how to verify that an integer is a solution to a linear equation.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Introduction to Algebraic Equations (L5.1)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/eBameNAndKw?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-9343\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 146556. <strong>Authored by<\/strong>: Lumen Learning. <strong>Provided by<\/strong>: `. <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>: MathAS Community License<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Introduction to Algebraic Equations (L5.1). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/eBameNAndKw\">https:\/\/youtu.be\/eBameNAndKw<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Introduction to Algebraic Equations (L5.1)\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/eBameNAndKw\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 146556\",\"author\":\"Lumen Learning\",\"organization\":\"`\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"MathAS Community 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