{"id":2034,"date":"2023-04-21T17:44:42","date_gmt":"2023-04-21T17:44:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2034"},"modified":"2024-10-18T20:55:42","modified_gmt":"2024-10-18T20:55:42","slug":"exponential-quadratic-and-logarithmic-functions-background-youll-need-page-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/exponential-quadratic-and-logarithmic-functions-background-youll-need-page-1\/","title":{"raw":"Personal Finance - Common Scenarios: Background You'll Need - Page 1","rendered":"Personal Finance &#8211; Common Scenarios: Background You&#8217;ll Need &#8211; Page 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Solve linear equations&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:7041,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:2,&quot;11&quot;:3,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Solve linear equations<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Linear Equations<\/h2>\r\n<p>Before we delve into solving linear equations, it's essential to grasp the basic principles behind them. A <strong>linear equation<\/strong> is an algebraic statement where two expressions are set equal to each other, often involving a constant and a variable. The goal is to find the value of the variable that makes the equation true. These equations are fundamental in algebra and provide a gateway to understanding more complex mathematical concepts.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How to: Solve a linear equation:<\/strong><\/p>\r\n<ol>\r\n\t<li>Begin by simplifying the expressions on each side of the equation as much as possible.<\/li>\r\n\t<li>Use inverse operations to isolate the variable on one side of the equation.<\/li>\r\n\t<li>Perform the same operations on both sides to maintain equality.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p>Remember, solving linear equations is not just about finding an answer\u2014it's about understanding the relationship between variables and constants, and how changes in one affect the other.<\/p>\r\n<section class=\"textbox proTip\">Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation first. You should always simplify as much as possible before trying to isolate the variable.<\/section>\r\n<section class=\"textbox example\">Solve:<br \/>\r\n<center>[latex]3x - 7 - 2x - 4=1[\/latex]<\/center><br \/>\r\n[reveal-answer q=\"963555\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"963555\"]The left side of the equation has an expression that we should simplify before trying to isolate the variable.\r\n\r\n<table id=\"eip-id1168467171254\" class=\"unnumbered unstyled\" summary=\"The first line shows the equation 3x minus 7 minus 2x minus 4 equals 1. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]3x-7-2x-4=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rearrange the terms, using the Commutative Property of Addition.<\/td>\r\n<td>[latex]3x-2x-7-4=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]x-11=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add [latex]11[\/latex] to both sides to isolate [latex]x[\/latex] .<\/td>\r\n<td>[latex]x-11(\\color{red}{+11}\\color{black})=1(\\color{red}{+11}\\color{black})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.<\/td>\r\n<td>Substitute [latex]x=12[\/latex] into the original equation.<br \/>\r\n[latex]\\begin{array}{rcl} 3x-7-2x-4 &amp; = &amp; 1 \\\\ 3(\\color{red}{12}\\color{black})-7-2(\\color{red}{12}\\color{black})-4 &amp; = &amp; 1 \\\\ 36-7-24-4 &amp; = &amp; 1 \\\\ 29-24-4 &amp; = &amp; 1 \\\\ 5-4 &amp; = &amp; 1 \\\\ 1 &amp; = &amp; 1 \\quad \\checkmark \\end{array} [\/latex]\r\n\r\n<p>&nbsp;<\/p>\r\n\r\nThe solution checks.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]4478[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Solve:<br \/>\r\n<center>[latex]2\\left(3k - 1\\right)-5k=-2 - 7[\/latex]<\/center>\r\n<p>[reveal-answer q=\"274755\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"274755\"]<\/p>\r\n<p>Both sides of the equation have expressions that we should simplify before we isolate the variable.<\/p>\r\n<table id=\"eip-id1168469785088\" class=\"unnumbered unstyled\" summary=\"The top line says 2 times parentheses 3k minus 1 minus 5k equals negative 2 minus 7. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]2(3k-1)-5k=-2-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute on the left, subtract on the right.<\/td>\r\n<td>[latex]6k-2-5k=-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property of Addition.<\/td>\r\n<td>[latex]6k-5k-2=-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]k-2=-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Undo subtraction by using the Addition Property of Equality.<\/td>\r\n<td>\r\n<p>[latex]k-2(\\color{red}{+2}\\color{black})=-9(\\color{red}{+2}\\color{black})[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]k=-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.<\/td>\r\n<td>Let [latex]k=-7[\/latex].<br \/>\r\n[latex]\\begin{array}{rcl} 2(3k-1)-5k &amp; = &amp; -2-7 \\\\ 2\\left(3(\\color{red}{-7}\\color{black})-1\\right)-5(\\color{red}{-7}\\color{black}) &amp; = &amp; -2-7 \\\\ 2(-21-1)-5(-7) &amp; = &amp; -9 \\\\ 2(-22)+35 &amp; = &amp; -9 \\\\ -44+35 &amp; = &amp; -9 \\\\ -9 &amp; = &amp; -9 \\quad \\checkmark \\end{array} [\/latex]\r\n\r\n<p>&nbsp;<\/p>\r\n\r\nThe solution checks.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p class=\"p1\">[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]4481[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Solve linear equations&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:7041,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:2,&quot;11&quot;:3,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Solve linear equations<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Solving Linear Equations<\/h2>\n<p>Before we delve into solving linear equations, it&#8217;s essential to grasp the basic principles behind them. A <strong>linear equation<\/strong> is an algebraic statement where two expressions are set equal to each other, often involving a constant and a variable. The goal is to find the value of the variable that makes the equation true. These equations are fundamental in algebra and provide a gateway to understanding more complex mathematical concepts.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: Solve a linear equation:<\/strong><\/p>\n<ol>\n<li>Begin by simplifying the expressions on each side of the equation as much as possible.<\/li>\n<li>Use inverse operations to isolate the variable on one side of the equation.<\/li>\n<li>Perform the same operations on both sides to maintain equality.<\/li>\n<\/ol>\n<\/section>\n<p>Remember, solving linear equations is not just about finding an answer\u2014it&#8217;s about understanding the relationship between variables and constants, and how changes in one affect the other.<\/p>\n<section class=\"textbox proTip\">Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation first. You should always simplify as much as possible before trying to isolate the variable.<\/section>\n<section class=\"textbox example\">Solve:<\/p>\n<div style=\"text-align: center;\">[latex]3x - 7 - 2x - 4=1[\/latex]<\/div>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q963555\">Show Answer<\/button><\/p>\n<div id=\"q963555\" class=\"hidden-answer\" style=\"display: none\">The left side of the equation has an expression that we should simplify before trying to isolate the variable.<\/p>\n<table id=\"eip-id1168467171254\" class=\"unnumbered unstyled\" summary=\"The first line shows the equation 3x minus 7 minus 2x minus 4 equals 1. The next line says,\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]3x-7-2x-4=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rearrange the terms, using the Commutative Property of Addition.<\/td>\n<td>[latex]3x-2x-7-4=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]x-11=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add [latex]11[\/latex] to both sides to isolate [latex]x[\/latex] .<\/td>\n<td>[latex]x-11(\\color{red}{+11}\\color{black})=1(\\color{red}{+11}\\color{black})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x=12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.<\/td>\n<td>Substitute [latex]x=12[\/latex] into the original equation.<br \/>\n[latex]\\begin{array}{rcl} 3x-7-2x-4 & = & 1 \\\\ 3(\\color{red}{12}\\color{black})-7-2(\\color{red}{12}\\color{black})-4 & = & 1 \\\\ 36-7-24-4 & = & 1 \\\\ 29-24-4 & = & 1 \\\\ 5-4 & = & 1 \\\\ 1 & = & 1 \\quad \\checkmark \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The solution checks.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm4478\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=4478&theme=lumen&iframe_resize_id=ohm4478&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Solve:<\/p>\n<div style=\"text-align: center;\">[latex]2\\left(3k - 1\\right)-5k=-2 - 7[\/latex]<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q274755\">Show Answer<\/button><\/p>\n<div id=\"q274755\" class=\"hidden-answer\" style=\"display: none\">\n<p>Both sides of the equation have expressions that we should simplify before we isolate the variable.<\/p>\n<table id=\"eip-id1168469785088\" class=\"unnumbered unstyled\" summary=\"The top line says 2 times parentheses 3k minus 1 minus 5k equals negative 2 minus 7. The next line says\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]2(3k-1)-5k=-2-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute on the left, subtract on the right.<\/td>\n<td>[latex]6k-2-5k=-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property of Addition.<\/td>\n<td>[latex]6k-5k-2=-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]k-2=-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Undo subtraction by using the Addition Property of Equality.<\/td>\n<td>\n[latex]k-2(\\color{red}{+2}\\color{black})=-9(\\color{red}{+2}\\color{black})[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]k=-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.<\/td>\n<td>Let [latex]k=-7[\/latex].<br \/>\n[latex]\\begin{array}{rcl} 2(3k-1)-5k & = & -2-7 \\\\ 2\\left(3(\\color{red}{-7}\\color{black})-1\\right)-5(\\color{red}{-7}\\color{black}) & = & -2-7 \\\\ 2(-21-1)-5(-7) & = & -9 \\\\ 2(-22)+35 & = & -9 \\\\ -44+35 & = & -9 \\\\ -9 & = & -9 \\quad \\checkmark \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The solution checks.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"p1\"><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm4481\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=4481&theme=lumen&iframe_resize_id=ohm4481&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":2,"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\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":4885,"module-header":"background_you_need","content_attributions":[{"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"}],"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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2034"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/16"}],"version-history":[{"count":20,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2034\/revisions"}],"predecessor-version":[{"id":13102,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2034\/revisions\/13102"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/4885"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2034\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2034"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2034"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2034"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2034"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}