{"id":9409,"date":"2017-05-02T22:20:15","date_gmt":"2017-05-02T22:20:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9409"},"modified":"2024-04-29T18:47:14","modified_gmt":"2024-04-29T18:47:14","slug":"determining-whether-a-whole-number-is-a-solution-to-an-equation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/determining-whether-a-whole-number-is-a-solution-to-an-equation\/","title":{"raw":"Determining Whether a Whole Number is a Solution to an Equation","rendered":"Determining Whether a Whole Number 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><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Determine whether a number is a solution of an equation&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6529,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:4,&quot;14&quot;:[null,2,0],&quot;15&quot;:&quot;Calibri&quot;}\">Determine whether a number is a solution of an equation<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Determine Whether a Number is a Solution of an Equation<\/h2>\r\nSolving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable that make the equation a true statement. Any number that makes the equation true is called a solution of the equation. It is the answer to the puzzle!\r\n<div class=\"textbox shaded\">\r\n<h3>Solution of an Equation<\/h3>\r\nA solution to an equation is a value of a variable that makes a true statement when substituted into the equation.\r\nThe process of finding the solution to an equation is called solving the equation.\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nTo find the solution to an equation means to find the value of the variable that makes the equation true. Can you recognize the solution of [latex]x+2=7?[\/latex] If you said [latex]5[\/latex], you\u2019re right! We say [latex]5[\/latex] is a solution to the equation [latex]x+2=7[\/latex] because when we substitute [latex]5[\/latex] for [latex]x[\/latex] the resulting statement is true.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\hfill x+2=7\\hfill \\\\ \\hfill 5+2\\stackrel{?}{=}7\\hfill \\\\ \\\\ \\hfill 7=7\\quad\\checkmark \\hfill \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Since [latex]5+2=7[\/latex] is a true statement, we know that [latex]5[\/latex] is indeed a solution to the equation.<\/p>\r\n<p style=\"text-align: left;\">The symbol [latex]\\stackrel{?}{=}[\/latex] asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign [latex]=[\/latex] or not-equal sign [latex]\\not=[\/latex].<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Determine whether a number is a solution to an equation.<\/h3>\r\n<ol id=\"eip-id1168468428753\" 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=\"eip-409\">\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 [latex]x=5[\/latex] is a solution of [latex]6x - 17=16[\/latex].\r\n\r\nSolution\r\n<table id=\"eip-id1168469790662\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation 6 x minus 17 equal to 16. Substitute 5 for x and the equation becomes 6 times 5 minus 17 equal to 16. Is this true? Simplify the left side of the equation by multiplying 6 by 5 to get 30. The left side becomes 30 minus 17 which is 13. Thirteen is not equal to 16 on the right side of the equation.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]6x--17=16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{5}[\/latex] for x.<\/td>\r\n<td>[latex]6\\cdot\\color{red}{5}--17=16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]30--17=16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]13\\not=16[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo [latex]x=5[\/latex] is not a solution to the equation [latex]6x - 17=16[\/latex].\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]146455[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDetermine whether [latex]y=2[\/latex] is a solution of [latex]6y - 4=5y - 2[\/latex].\r\n[reveal-answer q=\"23524\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"23524\"]\r\n\r\nSolution\r\nHere, the variable appears on both sides of the equation. We must substitute [latex]2[\/latex] for each [latex]y[\/latex].\r\n<table id=\"eip-id1168469647895\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation 6 y minus 4 equal to 5 y minus 2. Substitute 2 for y on both sides of the equation. The equation becomes 6 times 2 minus 4 equal to 5 times 2 minus 2. Is this true? Simplify the left side of the equation by multiplying 6 by 2 to get 12. Then subtract 4 from 12 to get 8. Simplify the right side of the equation by multiplying 5 by 2 to get 10. Then subtract 2 from 10 to get eight. Both sides of the equation are 8.\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]6y--4=5y--2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{2}[\/latex] for y.<\/td>\r\n<td>[latex]6(\\color{red}{2})--4=5(\\color{red}{2})--2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]12--4=10--2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]8=8\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince [latex]y=2[\/latex] results in a true equation, we know that [latex]2[\/latex] is a solution to the equation [latex]6y - 4=5y - 2[\/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]146456[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to verify whether an integer is a solution to a linear equation.\r\n\r\nhttps:\/\/youtu.be\/eBameNAndKw","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Determine whether a number is a solution of an equation&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6529,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:4,&quot;14&quot;:[null,2,0],&quot;15&quot;:&quot;Calibri&quot;}\">Determine whether a number is a solution of an equation<\/span><\/li>\n<\/ul>\n<\/div>\n<h2>Determine Whether a Number is a Solution of an Equation<\/h2>\n<p>Solving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable that make the equation a true statement. Any number that makes the equation true is called a solution of the equation. It is the answer to the puzzle!<\/p>\n<div class=\"textbox shaded\">\n<h3>Solution of an Equation<\/h3>\n<p>A solution to an equation is a value of a variable that makes a true statement when substituted into the equation.<br \/>\nThe process of finding the solution to an equation is called solving the equation.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>To find the solution to an equation means to find the value of the variable that makes the equation true. Can you recognize the solution of [latex]x+2=7?[\/latex] If you said [latex]5[\/latex], you\u2019re right! We say [latex]5[\/latex] is a solution to the equation [latex]x+2=7[\/latex] because when we substitute [latex]5[\/latex] for [latex]x[\/latex] the resulting statement is true.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ \\hfill x+2=7\\hfill \\\\ \\hfill 5+2\\stackrel{?}{=}7\\hfill \\\\ \\\\ \\hfill 7=7\\quad\\checkmark \\hfill \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Since [latex]5+2=7[\/latex] is a true statement, we know that [latex]5[\/latex] is indeed a solution to the equation.<\/p>\n<p style=\"text-align: left;\">The symbol [latex]\\stackrel{?}{=}[\/latex] asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign [latex]=[\/latex] or not-equal sign [latex]\\not=[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Determine whether a number is a solution to an equation.<\/h3>\n<ol id=\"eip-id1168468428753\" 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=\"eip-409\">\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 [latex]x=5[\/latex] is a solution of [latex]6x - 17=16[\/latex].<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168469790662\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation 6 x minus 17 equal to 16. Substitute 5 for x and the equation becomes 6 times 5 minus 17 equal to 16. Is this true? Simplify the left side of the equation by multiplying 6 by 5 to get 30. The left side becomes 30 minus 17 which is 13. Thirteen is not equal to 16 on the right side of the equation.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]6x--17=16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{5}[\/latex] for x.<\/td>\n<td>[latex]6\\cdot\\color{red}{5}--17=16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]30--17=16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]13\\not=16[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So [latex]x=5[\/latex] is not a solution to the equation [latex]6x - 17=16[\/latex].<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146455\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146455&theme=oea&iframe_resize_id=ohm146455&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>Determine whether [latex]y=2[\/latex] is a solution of [latex]6y - 4=5y - 2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q23524\">Show Solution<\/span><\/p>\n<div id=\"q23524\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nHere, the variable appears on both sides of the equation. We must substitute [latex]2[\/latex] for each [latex]y[\/latex].<\/p>\n<table id=\"eip-id1168469647895\" class=\"unnumbered unstyled\" summary=\"The image shows the given equation 6 y minus 4 equal to 5 y minus 2. Substitute 2 for y on both sides of the equation. The equation becomes 6 times 2 minus 4 equal to 5 times 2 minus 2. Is this true? Simplify the left side of the equation by multiplying 6 by 2 to get 12. Then subtract 4 from 12 to get 8. Simplify the right side of the equation by multiplying 5 by 2 to get 10. Then subtract 2 from 10 to get eight. Both sides of the equation are 8.\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]6y--4=5y--2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{2}[\/latex] for y.<\/td>\n<td>[latex]6(\\color{red}{2})--4=5(\\color{red}{2})--2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]12--4=10--2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]8=8\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since [latex]y=2[\/latex] results in a true equation, we know that [latex]2[\/latex] is a solution to the equation [latex]6y - 4=5y - 2[\/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=\"ohm146456\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146456&theme=oea&iframe_resize_id=ohm146456&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to verify whether an integer is a solution to a linear equation.<\/p>\n<p><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-9409\">\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 146456, 146455. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License<\/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\/about\/pdm\">Public Domain: No Known Copyright<\/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":13,"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\":\"pd\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 146456, 146455\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License\"}]","CANDELA_OUTCOMES_GUID":"08370cb3fefd4be3ac2330c002b25e3f, d2175246540548558711ad92a345f318","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-9409","chapter","type-chapter","status-publish","hentry"],"part":13769,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9409","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":17,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9409\/revisions"}],"predecessor-version":[{"id":17115,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9409\/revisions\/17115"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/13769"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9409\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=9409"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=9409"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=9409"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=9409"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}