{"id":2579,"date":"2025-08-13T18:07:05","date_gmt":"2025-08-13T18:07:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2579"},"modified":"2026-01-13T17:45:12","modified_gmt":"2026-01-13T17:45:12","slug":"polynomial-equations-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polynomial-equations-background-youll-need-3\/","title":{"raw":"Polynomial Equations: Background You'll Need 3","rendered":"Polynomial Equations: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Solve linear equations<\/span><\/li>\r\n<\/ul>\r\n<\/section>Solving linear equations involves the fundamental properties of equality and basic algebraic operations. Some equations can be solved quickly in your head. For example, what is the value of [latex]y[\/latex] in the equation [latex]2y=6[\/latex]? You can easily determine that [latex]y=3[\/latex] by dividing both sides by [latex]2[\/latex].\r\n\r\nOther equations are more complicated. Solving [latex]\\displaystyle 4\\left(\\frac{1}{3}\\normalsize t+\\frac{1}{2}\\normalsize\\right)=6[\/latex] without writing anything down is difficult! That is because this equation contains not just a variable but also fractions and terms inside parentheses. This is a <b>multi-step equation,\u00a0<\/b>one that takes several steps to solve. Although multi-step equations take more time and more operations, they can still be simplified and solved by applying basic algebraic rules.\r\n\r\nRemember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The [pb_glossary id=\"13469\"]addition property of equality[\/pb_glossary] and the [pb_glossary id=\"13470\"]multiplication property of equality[\/pb_glossary] explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you will keep both sides of the equation equal.\r\n\r\n<section class=\"textbox questionHelp\"><strong>How To: Solving Multi-Step Equations<\/strong>\r\n<ol>\r\n \t<li>(Optional) Multiply to clear any fractions or decimals.<\/li>\r\n \t<li>Simplify each side by clearing parentheses and combining like terms.<\/li>\r\n \t<li>Add or subtract to isolate the variable term\u2014you may have to move a term with the variable.<\/li>\r\n \t<li>Multiply or divide to isolate the variable.<\/li>\r\n \t<li>Check the solution.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<center>[latex]3x+5x+4-x+7=88[\/latex]<\/center>[reveal-answer q=\"455516\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"455516\"]There are three like terms involving a variable: [latex]3x[\/latex], [latex]5x[\/latex], and [latex]\u2013x[\/latex]. Combine these like terms. [latex]4[\/latex] and [latex]7[\/latex] are also like terms and can be added.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,3x+5x+4-x+7=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x+11=\\,\\,\\,88\\end{array}[\/latex]<\/p>\r\nThe equation is now in the form [latex]ax+b=c[\/latex], so we can solve as before.\r\n\r\nSubtract [latex]11[\/latex] from both sides.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}7x+11\\,\\,\\,=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-11\\,\\,\\,\\,\\,\\,\\,-11}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x\\,\\,\\,=\\,\\,\\,77\\end{array}[\/latex]<\/p>\r\nDivide both sides by [latex]7[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{7x}\\,\\,\\,=\\,\\,\\,\\underline{77}\\\\7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,=\\,\\,\\,11\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>Some equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[\/latex].\r\n\r\nTo solve this equation, we need to \u201cmove\u201d one of the variable terms. This can make it difficult to decide which side to work with. It does not matter which term gets moved, [latex]4x[\/latex] or [latex]2x[\/latex]; however, to avoid negative coefficients, you can move the smaller term.\r\n\r\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<center>[latex]4x-6=2x+10[\/latex]<\/center>[reveal-answer q=\"457216\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"457216\"]Choose the variable term to move\u2014to avoid negative terms choose [latex]2x[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x-6=2x+10\\,\\\\\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\2x-6=10\\end{array}[\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"text-align: left;\">Now add [latex]6[\/latex] to both sides to isolate the term with the variable.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x-6=10\\\\\\underline{\\,\\,\\,\\,+6\\,\\,\\,+6}\\\\2x=16\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Now divide each side by [latex]2[\/latex] to isolate the variable [latex]x[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\Large\\frac{2x}{2}\\normalsize=\\Large\\frac{16}{2}\\\\\\\\\\normalsize{x=8}\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]318847[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Solve linear equations<\/span><\/li>\n<\/ul>\n<\/section>\n<p>Solving linear equations involves the fundamental properties of equality and basic algebraic operations. Some equations can be solved quickly in your head. For example, what is the value of [latex]y[\/latex] in the equation [latex]2y=6[\/latex]? You can easily determine that [latex]y=3[\/latex] by dividing both sides by [latex]2[\/latex].<\/p>\n<p>Other equations are more complicated. Solving [latex]\\displaystyle 4\\left(\\frac{1}{3}\\normalsize t+\\frac{1}{2}\\normalsize\\right)=6[\/latex] without writing anything down is difficult! That is because this equation contains not just a variable but also fractions and terms inside parentheses. This is a <b>multi-step equation,\u00a0<\/b>one that takes several steps to solve. Although multi-step equations take more time and more operations, they can still be simplified and solved by applying basic algebraic rules.<\/p>\n<p>Remember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The addition property of equality and the multiplication property of equality explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you will keep both sides of the equation equal.<\/p>\n<section class=\"textbox questionHelp\"><strong>How To: Solving Multi-Step Equations<\/strong><\/p>\n<ol>\n<li>(Optional) Multiply to clear any fractions or decimals.<\/li>\n<li>Simplify each side by clearing parentheses and combining like terms.<\/li>\n<li>Add or subtract to isolate the variable term\u2014you may have to move a term with the variable.<\/li>\n<li>Multiply or divide to isolate the variable.<\/li>\n<li>Check the solution.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]3x+5x+4-x+7=88[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q455516\">Show Solution<\/button><\/p>\n<div id=\"q455516\" class=\"hidden-answer\" style=\"display: none\">There are three like terms involving a variable: [latex]3x[\/latex], [latex]5x[\/latex], and [latex]\u2013x[\/latex]. Combine these like terms. [latex]4[\/latex] and [latex]7[\/latex] are also like terms and can be added.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,3x+5x+4-x+7=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x+11=\\,\\,\\,88\\end{array}[\/latex]<\/p>\n<p>The equation is now in the form [latex]ax+b=c[\/latex], so we can solve as before.<\/p>\n<p>Subtract [latex]11[\/latex] from both sides.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}7x+11\\,\\,\\,=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-11\\,\\,\\,\\,\\,\\,\\,-11}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x\\,\\,\\,=\\,\\,\\,77\\end{array}[\/latex]<\/p>\n<p>Divide both sides by [latex]7[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{7x}\\,\\,\\,=\\,\\,\\,\\underline{77}\\\\7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,=\\,\\,\\,11\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Some equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[\/latex].<\/p>\n<p>To solve this equation, we need to \u201cmove\u201d one of the variable terms. This can make it difficult to decide which side to work with. It does not matter which term gets moved, [latex]4x[\/latex] or [latex]2x[\/latex]; however, to avoid negative coefficients, you can move the smaller term.<\/p>\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]4x-6=2x+10[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q457216\">Show Solution<\/button><\/p>\n<div id=\"q457216\" class=\"hidden-answer\" style=\"display: none\">Choose the variable term to move\u2014to avoid negative terms choose [latex]2x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x-6=2x+10\\,\\\\\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\2x-6=10\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">Now add [latex]6[\/latex] to both sides to isolate the term with the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x-6=10\\\\\\underline{\\,\\,\\,\\,+6\\,\\,\\,+6}\\\\2x=16\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Now divide each side by [latex]2[\/latex] to isolate the variable [latex]x[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\Large\\frac{2x}{2}\\normalsize=\\Large\\frac{16}{2}\\\\\\\\\\normalsize{x=8}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm318847\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318847&theme=lumen&iframe_resize_id=ohm318847&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_2579_13469\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_2579_13469\"><div tabindex=\"-1\"><\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_2579_13470\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_2579_13470\"><div tabindex=\"-1\"><\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":67,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":506,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2579"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2579\/revisions"}],"predecessor-version":[{"id":5320,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2579\/revisions\/5320"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/506"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2579\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2579"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2579"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2579"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2579"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}