{"id":444,"date":"2024-04-19T00:43:35","date_gmt":"2024-04-19T00:43:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=444"},"modified":"2024-11-20T00:51:34","modified_gmt":"2024-11-20T00:51:34","slug":"algebra-essentials-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/algebra-essentials-background-youll-need-3\/","title":{"raw":"Algebra Essentials: Background You'll Need 3","rendered":"Algebra Essentials: Background You&#8217;ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Learn to spot the difference between letters and numbers in math expressions and combine similar terms to simplify them.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Simplifying Algebraic Expressions<\/h2>\r\nAlgebraic expressions are combinations of variables, constants, and mathematical operations. Simplifying these expressions is a fundamental skill in algebra that helps us solve equations more easily and understand mathematical relationships better.\r\n\r\nBefore we begin simplifying, it's crucial to understand the difference between variables and constants.\r\n<h3>Constant and Variables<\/h3>\r\nIn the study of mathematics and programming, two fundamental concepts that frequently arise are <strong>constants<\/strong> and <strong>variables<\/strong>. [pb_glossary id=\"446\"]Constants[\/pb_glossary] are elements that remain unchanged within a given scenario, providing a stable and known value that can simplify calculations and coding. On the other hand, [pb_glossary id=\"447\"]variables[\/pb_glossary] represent elements whose values can change, depending on the conditions of the problem or the inputs to a program. These concepts are crucial for developing equations, functions, and algorithms that accurately model real-world phenomena. Understanding the differences between constants and variables is key to mastering mathematical expressions and enhancing problem-solving skills.\r\n<table style=\"border-collapse: collapse; width: 100%; height: 88px;\">\r\n<tbody>\r\n<tr style=\"height: 22px;\">\r\n<th style=\"width: 16.4159%; height: 22px;\"><strong>Aspect<\/strong><\/th>\r\n<th style=\"width: 41.4501%; height: 22px;\"><strong>Constant<\/strong><\/th>\r\n<th style=\"width: 42.1341%; height: 22px;\"><strong>Variable<\/strong><\/th>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 16.4159%; height: 22px;\"><strong>Value Stability<\/strong><\/td>\r\n<td style=\"width: 41.4501%; height: 22px;\">Remain fixed and unchanging.<\/td>\r\n<td style=\"width: 42.1341%; height: 22px;\">Can change based on context or conditions.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 16.4159%; height: 22px;\"><strong>Representation<\/strong><\/td>\r\n<td style=\"width: 41.4501%; height: 22px;\">Fixed numerical values (e.g., [latex]5, -3, \\frac{1}{2}[\/latex])<\/td>\r\n<td style=\"width: 42.1341%; height: 22px;\">Letters or symbols that represent unknown or changing values (e.g., [latex]x, y, z[\/latex])<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 16.4159%; height: 22px;\"><strong>Role<\/strong><\/td>\r\n<td style=\"width: 41.4501%; height: 22px;\">Known values in calculations and programming.<\/td>\r\n<td style=\"width: 42.1341%; height: 22px;\">Unknown or variable quantities to be determined.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox example\">Identify the constant and variables of the following algebraic expressions.\r\n<ul>\r\n \t<li>[latex]3x+4[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"327081\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"327081\"]Here, [latex]x[\/latex] is the variable as it represents an unknown value that can change. The number [latex]4[\/latex] is the constant because it is a specific, unchanging value added to the variable part.\r\nNote: [latex]3[\/latex] is the coefficient of the variable [latex]x[\/latex]. It multiplies the variable. However, when discussing algebraic expressions, the term \"constant\" typically refers to standalone values, not coefficients.[\/hidden-answer]\r\n<ul>\r\n \t<li>[latex]x^2 +3x+4[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"371956\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"371956\"]Here, [latex]x[\/latex] is the variable, appearing in two terms with different powers. The number [latex]4[\/latex] is the constant because it is a specific, unchanging value added to the variable part. [\/hidden-answer]\r\n<ul>\r\n \t<li>[latex]2xy[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"270020\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"270020\"]\r\n<ul>\r\n \t<li>Variables: [latex]x[\/latex] and [latex]y[\/latex] are both variables. They represent quantities that can vary.<\/li>\r\n \t<li>Coefficient: [latex]2[\/latex] is the coefficient. It multiplies the product of two variables, [latex]x[\/latex] and [latex]y[\/latex].<\/li>\r\n \t<li>There is no constant term in this algebraic expression.[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3>Combining Like Terms<\/h3>\r\nCombining like terms is an essential technique in simplifying algebraic expressions. To combine like terms, first identify terms within the expression that have identical variables raised to the same power. Once identified, you can add or subtract their coefficients while keeping the variable part unchanged.\r\n\r\n<strong>Characteristics of Like Terms:<\/strong>\r\n<ul>\r\n \t<li><strong>Same Variables:<\/strong> Like terms must involve the exact same variables. For example, [latex]2x[\/latex] and [latex]5x[\/latex] are like terms because both contain the variable [latex]x[\/latex].<\/li>\r\n \t<li><strong>Same Exponents:<\/strong> The variables in like terms must be raised to the same power. For instance, [latex]3x^2[\/latex] and [latex]-7x^2 [\/latex] are like terms because both terms include [latex]x[\/latex] squared.<\/li>\r\n \t<li><strong>Coefficients Can Vary:<\/strong> The coefficients (numerical values multiplying the variables) can be different. In the examples above, [latex]2[\/latex] and [latex]5[\/latex] are different coefficients for terms involving [latex]x[\/latex], and [latex]3[\/latex] and [latex]-7[\/latex] are different coefficients for terms involving [latex]x^2[\/latex].<\/li>\r\n<\/ul>\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"font-600 text-xl font-bold\"><strong>How to: Simplify Algebraic Expressions by Combining Like Terms<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify variables and constants in the expression.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Group like terms together.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine like terms by adding or subtracting their coefficients.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write the simplified expression with unlike terms separated.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Simplify the following algebraic expression:<center>[latex]5x-2y-8x+7y[\/latex]<\/center>[reveal-answer q=\"730653\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"730653\"]The like terms in this expression are:\r\n<ul>\r\n \t<li style=\"text-align: left;\">[latex]5x[\/latex] and [latex]-8x[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">[latex]-2y[\/latex] and [latex]7y[\/latex]<\/li>\r\n<\/ul>\r\n<p style=\"text-align: left;\">Note how we kept the sign in front of each term.<\/p>\r\n<p style=\"text-align: left;\">Combine like terms:<\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: left;\">[latex]5x-8x = (5-8)x = -3x[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">[latex]-2y+7y = (-2+7)y = 5y[\/latex]<\/li>\r\n<\/ul>\r\n<p style=\"text-align: left;\">Note how signs become operations when you combine like terms.<\/p>\r\n<p style=\"text-align: left;\">Simplified Expression:<\/p>\r\n<p style=\"text-align: center;\">[latex]5x-2y-8x+7y=-3x+5y[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18654[\/ohm2_question]<\/section><section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18655[\/ohm2_question]<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Learn to spot the difference between letters and numbers in math expressions and combine similar terms to simplify them.<\/li>\n<\/ul>\n<\/section>\n<h2>Simplifying Algebraic Expressions<\/h2>\n<p>Algebraic expressions are combinations of variables, constants, and mathematical operations. Simplifying these expressions is a fundamental skill in algebra that helps us solve equations more easily and understand mathematical relationships better.<\/p>\n<p>Before we begin simplifying, it&#8217;s crucial to understand the difference between variables and constants.<\/p>\n<h3>Constant and Variables<\/h3>\n<p>In the study of mathematics and programming, two fundamental concepts that frequently arise are <strong>constants<\/strong> and <strong>variables<\/strong>. <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_444_446\">Constants<\/a> are elements that remain unchanged within a given scenario, providing a stable and known value that can simplify calculations and coding. On the other hand, <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_444_447\">variables<\/a> represent elements whose values can change, depending on the conditions of the problem or the inputs to a program. These concepts are crucial for developing equations, functions, and algorithms that accurately model real-world phenomena. Understanding the differences between constants and variables is key to mastering mathematical expressions and enhancing problem-solving skills.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 88px;\">\n<tbody>\n<tr style=\"height: 22px;\">\n<th style=\"width: 16.4159%; height: 22px;\"><strong>Aspect<\/strong><\/th>\n<th style=\"width: 41.4501%; height: 22px;\"><strong>Constant<\/strong><\/th>\n<th style=\"width: 42.1341%; height: 22px;\"><strong>Variable<\/strong><\/th>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 16.4159%; height: 22px;\"><strong>Value Stability<\/strong><\/td>\n<td style=\"width: 41.4501%; height: 22px;\">Remain fixed and unchanging.<\/td>\n<td style=\"width: 42.1341%; height: 22px;\">Can change based on context or conditions.<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 16.4159%; height: 22px;\"><strong>Representation<\/strong><\/td>\n<td style=\"width: 41.4501%; height: 22px;\">Fixed numerical values (e.g., [latex]5, -3, \\frac{1}{2}[\/latex])<\/td>\n<td style=\"width: 42.1341%; height: 22px;\">Letters or symbols that represent unknown or changing values (e.g., [latex]x, y, z[\/latex])<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 16.4159%; height: 22px;\"><strong>Role<\/strong><\/td>\n<td style=\"width: 41.4501%; height: 22px;\">Known values in calculations and programming.<\/td>\n<td style=\"width: 42.1341%; height: 22px;\">Unknown or variable quantities to be determined.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox example\">Identify the constant and variables of the following algebraic expressions.<\/p>\n<ul>\n<li>[latex]3x+4[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q327081\">Show Answer<\/button><\/p>\n<div id=\"q327081\" class=\"hidden-answer\" style=\"display: none\">Here, [latex]x[\/latex] is the variable as it represents an unknown value that can change. The number [latex]4[\/latex] is the constant because it is a specific, unchanging value added to the variable part.<br \/>\nNote: [latex]3[\/latex] is the coefficient of the variable [latex]x[\/latex]. It multiplies the variable. However, when discussing algebraic expressions, the term &#8220;constant&#8221; typically refers to standalone values, not coefficients.<\/div>\n<\/div>\n<ul>\n<li>[latex]x^2 +3x+4[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q371956\">Show Answer<\/button><\/p>\n<div id=\"q371956\" class=\"hidden-answer\" style=\"display: none\">Here, [latex]x[\/latex] is the variable, appearing in two terms with different powers. The number [latex]4[\/latex] is the constant because it is a specific, unchanging value added to the variable part. <\/div>\n<\/div>\n<ul>\n<li>[latex]2xy[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q270020\">Show Answer<\/button><\/p>\n<div id=\"q270020\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Variables: [latex]x[\/latex] and [latex]y[\/latex] are both variables. They represent quantities that can vary.<\/li>\n<li>Coefficient: [latex]2[\/latex] is the coefficient. It multiplies the product of two variables, [latex]x[\/latex] and [latex]y[\/latex].<\/li>\n<li>There is no constant term in this algebraic expression.<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<h3>Combining Like Terms<\/h3>\n<p>Combining like terms is an essential technique in simplifying algebraic expressions. To combine like terms, first identify terms within the expression that have identical variables raised to the same power. Once identified, you can add or subtract their coefficients while keeping the variable part unchanged.<\/p>\n<p><strong>Characteristics of Like Terms:<\/strong><\/p>\n<ul>\n<li><strong>Same Variables:<\/strong> Like terms must involve the exact same variables. For example, [latex]2x[\/latex] and [latex]5x[\/latex] are like terms because both contain the variable [latex]x[\/latex].<\/li>\n<li><strong>Same Exponents:<\/strong> The variables in like terms must be raised to the same power. For instance, [latex]3x^2[\/latex] and [latex]-7x^2[\/latex] are like terms because both terms include [latex]x[\/latex] squared.<\/li>\n<li><strong>Coefficients Can Vary:<\/strong> The coefficients (numerical values multiplying the variables) can be different. In the examples above, [latex]2[\/latex] and [latex]5[\/latex] are different coefficients for terms involving [latex]x[\/latex], and [latex]3[\/latex] and [latex]-7[\/latex] are different coefficients for terms involving [latex]x^2[\/latex].<\/li>\n<\/ul>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"font-600 text-xl font-bold\"><strong>How to: Simplify Algebraic Expressions by Combining Like Terms<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify variables and constants in the expression.<\/li>\n<li class=\"whitespace-normal break-words\">Group like terms together.<\/li>\n<li class=\"whitespace-normal break-words\">Combine like terms by adding or subtracting their coefficients.<\/li>\n<li class=\"whitespace-normal break-words\">Write the simplified expression with unlike terms separated.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Simplify the following algebraic expression:<\/p>\n<div style=\"text-align: center;\">[latex]5x-2y-8x+7y[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q730653\">Show Solution<\/button><\/p>\n<div id=\"q730653\" class=\"hidden-answer\" style=\"display: none\">The like terms in this expression are:<\/p>\n<ul>\n<li style=\"text-align: left;\">[latex]5x[\/latex] and [latex]-8x[\/latex]<\/li>\n<li style=\"text-align: left;\">[latex]-2y[\/latex] and [latex]7y[\/latex]<\/li>\n<\/ul>\n<p style=\"text-align: left;\">Note how we kept the sign in front of each term.<\/p>\n<p style=\"text-align: left;\">Combine like terms:<\/p>\n<ul>\n<li style=\"text-align: left;\">[latex]5x-8x = (5-8)x = -3x[\/latex]<\/li>\n<li style=\"text-align: left;\">[latex]-2y+7y = (-2+7)y = 5y[\/latex]<\/li>\n<\/ul>\n<p style=\"text-align: left;\">Note how signs become operations when you combine like terms.<\/p>\n<p style=\"text-align: left;\">Simplified Expression:<\/p>\n<p style=\"text-align: center;\">[latex]5x-2y-8x+7y=-3x+5y[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18654\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18654&theme=lumen&iframe_resize_id=ohm18654&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18655\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18655&theme=lumen&iframe_resize_id=ohm18655&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_444_446\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_444_446\"><div tabindex=\"-1\"><p>A constant is a specific, unchanging value used in algebraic expressions and equations. It remains the same regardless of the conditions of the problem.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_444_447\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_444_447\"><div tabindex=\"-1\"><p>A variable in mathematics is a symbol, typically a letter, used to represent an unknown or changeable value within an equation or expression.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":12,"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":32,"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/444"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/444\/revisions"}],"predecessor-version":[{"id":3279,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/444\/revisions\/3279"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/32"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/444\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=444"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=444"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=444"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=444"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}