{"id":3091,"date":"2019-10-23T14:06:03","date_gmt":"2019-10-23T14:06:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3091"},"modified":"2021-02-05T23:55:52","modified_gmt":"2021-02-05T23:55:52","slug":"summary-review-topics-5","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/summary-review-topics-5\/","title":{"raw":"Summary: Review Topics","rendered":"Summary: Review Topics"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li><strong>How to determine whether a number is a solution to an equation.<\/strong>\r\n<ul>\r\n \t<li>Step 1. Substitute the number for the variable in the equation.<\/li>\r\n \t<li>Step 2. Simplify the expressions on both sides of the equation.<\/li>\r\n \t<li>Step 3. Determine whether the resulting equation is true.\r\n<ul>\r\n \t<li style=\"list-style-type: none\">\r\n<ul>\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<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li><strong>Translate a word sentence to an algebraic equation.<\/strong>\r\n<ol id=\"eip-id1170324011027\" class=\"stepwise\">\r\n \t<li>Locate the \"equals\" word(s). Translate to an equal sign.<\/li>\r\n \t<li>Translate the words to the left of the \"equals\" word(s) into an algebraic expression.<\/li>\r\n \t<li>Translate the words to the right of the \"equals\" word(s) into an algebraic expression.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li><strong>Properties of Equalities<\/strong>\r\n<table id=\"fs-id908853\" class=\"unnumbered\" style=\"width: 85%\" summary=\"Table with two columns. The first column says \">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>Subtraction Property of Equality:<\/strong>\r\n\r\nFor any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex]\r\n\r\nif [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex].<\/td>\r\n<td><strong>Addition Property of Equality:<\/strong>\r\n\r\nFor any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex]\r\n\r\nif [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>Division Property of Equality:<\/strong>\r\n\r\nFor any numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex] where [latex]\\mathit{\\text{c}}\\ne \\mathit{0}[\/latex]\r\n\r\nif [latex]a=b[\/latex], then [latex]\r\n\r\n\\Large\\frac{a}{c}=\r\n\r\n\\Large\\frac{b}{c}[\/latex]<\/td>\r\n<td><strong>Multiplication Property of Equality:<\/strong>\r\n\r\nFor any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c}}[\/latex]\r\n\r\nif [latex]a=b[\/latex], then [latex]ac=bc[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li><strong>Summary of Fraction Operations<\/strong>\r\n<ul id=\"eip-id1170322929638\">\r\n \t<li><strong>Fraction multiplication:<\/strong> Multiply the numerators and multiply the denominators.\r\n[latex]\\Large\\frac{a}{b}\\cdot\\Large\\frac{c}{d}=\\Large\\frac{ac}{bd}[\/latex]<\/li>\r\n \t<li><strong>Fraction division:<\/strong> Multiply the first fraction by the reciprocal of the second.\r\n[latex]\\Large\\frac{a}{b}+\\Large\\frac{c}{d}=\\Large\\frac{a}{b}\\cdot\\Large\\frac{d}{c}[\/latex]<\/li>\r\n \t<li><strong>Fraction addition:<\/strong> Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.\r\n[latex]\\Large\\frac{a}{c}+\\Large\\frac{b}{c}=\\Large\\frac{a+b}{c}[\/latex]<\/li>\r\n \t<li><strong>Fraction subtraction:<\/strong> Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.\r\n[latex]\\Large\\frac{a}{c}-\\Large\\frac{b}{c}=\\Large\\frac{a-b}{c}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Simplify complex fractions.<\/strong>\r\n<ol id=\"eip-id1170321558052\" class=\"stepwise\">\r\n \t<li>Simplify the numerator.<\/li>\r\n \t<li>Simplify the denominator.<\/li>\r\n \t<li>Divide the numerator by the denominator.<\/li>\r\n \t<li>Simplify if possible.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<ul id=\"eip-809\">\r\n \t<li style=\"list-style-type: none\"><\/li>\r\n<\/ul>\r\n<ul id=\"eip-578\">\r\n \t<li><strong>Solve equations with fraction coefficients by clearing the fractions.<\/strong>\r\n<ol id=\"eip-id1170323900082\" class=\"stepwise\">\r\n \t<li>Find the least common denominator of <em>all<\/em> the fractions in the equation.<\/li>\r\n \t<li>Multiply both sides of the equation by that LCD. This clears the fractions.<\/li>\r\n \t<li>Solve using the General Strategy for Solving Linear Equations.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong>Solve an equation with variables and constants on both sides<\/strong>\r\n<ol id=\"eip-id1170322988380\" class=\"stepwise\">\r\n \t<li>Choose one side to be the variable side and then the other will be the constant side.<\/li>\r\n \t<li>Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.<\/li>\r\n \t<li>Collect the constants to the other side, using the Addition or Subtraction Property of Equality.<\/li>\r\n \t<li>Make the coefficient of the variable[latex]1[\/latex], using the Multiplication or Division Property of Equality.<\/li>\r\n \t<li>Check the solution by substituting into the original equation.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong>General strategy for solving linear equations<\/strong>\r\n<ol id=\"eip-id1170324010914\" class=\"stepwise\">\r\n \t<li>Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.<\/li>\r\n \t<li>Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.<\/li>\r\n \t<li>Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.<\/li>\r\n \t<li>Make the coefficient of the variable term equal to [latex]1[\/latex]. Use the Multiplication or Division Property of Equality. State the solution to the equation.<\/li>\r\n \t<li>Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li><strong>How to determine whether a number is a solution to an equation.<\/strong>\n<ul>\n<li>Step 1. Substitute the number for the variable in the equation.<\/li>\n<li>Step 2. Simplify the expressions on both sides of the equation.<\/li>\n<li>Step 3. Determine whether the resulting equation is true.\n<ul>\n<li style=\"list-style-type: none\">\n<ul>\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<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul>\n<li><strong>Translate a word sentence to an algebraic equation.<\/strong>\n<ol id=\"eip-id1170324011027\" class=\"stepwise\">\n<li>Locate the &#8220;equals&#8221; word(s). Translate to an equal sign.<\/li>\n<li>Translate the words to the left of the &#8220;equals&#8221; word(s) into an algebraic expression.<\/li>\n<li>Translate the words to the right of the &#8220;equals&#8221; word(s) into an algebraic expression.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<ul>\n<li><strong>Properties of Equalities<\/strong><br \/>\n<table id=\"fs-id908853\" class=\"unnumbered\" style=\"width: 85%\" summary=\"Table with two columns. The first column says\">\n<tbody>\n<tr valign=\"top\">\n<td><strong>Subtraction Property of Equality:<\/strong><\/p>\n<p>For any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex]<\/p>\n<p>if [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex].<\/td>\n<td><strong>Addition Property of Equality:<\/strong><\/p>\n<p>For any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex]<\/p>\n<p>if [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>Division Property of Equality:<\/strong><\/p>\n<p>For any numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c,}}[\/latex] where [latex]\\mathit{\\text{c}}\\ne \\mathit{0}[\/latex]<\/p>\n<p>if [latex]a=b[\/latex], then [latex]\\Large\\frac{a}{c}=    \\Large\\frac{b}{c}[\/latex]<\/td>\n<td><strong>Multiplication Property of Equality:<\/strong><\/p>\n<p>For any real numbers [latex]\\mathit{\\text{a, b,}}[\/latex] and [latex]\\mathit{\\text{c}}[\/latex]<\/p>\n<p>if [latex]a=b[\/latex], then [latex]ac=bc[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li><strong>Summary of Fraction Operations<\/strong>\n<ul id=\"eip-id1170322929638\">\n<li><strong>Fraction multiplication:<\/strong> Multiply the numerators and multiply the denominators.<br \/>\n[latex]\\Large\\frac{a}{b}\\cdot\\Large\\frac{c}{d}=\\Large\\frac{ac}{bd}[\/latex]<\/li>\n<li><strong>Fraction division:<\/strong> Multiply the first fraction by the reciprocal of the second.<br \/>\n[latex]\\Large\\frac{a}{b}+\\Large\\frac{c}{d}=\\Large\\frac{a}{b}\\cdot\\Large\\frac{d}{c}[\/latex]<\/li>\n<li><strong>Fraction addition:<\/strong> Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.<br \/>\n[latex]\\Large\\frac{a}{c}+\\Large\\frac{b}{c}=\\Large\\frac{a+b}{c}[\/latex]<\/li>\n<li><strong>Fraction subtraction:<\/strong> Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.<br \/>\n[latex]\\Large\\frac{a}{c}-\\Large\\frac{b}{c}=\\Large\\frac{a-b}{c}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Simplify complex fractions.<\/strong>\n<ol id=\"eip-id1170321558052\" class=\"stepwise\">\n<li>Simplify the numerator.<\/li>\n<li>Simplify the denominator.<\/li>\n<li>Divide the numerator by the denominator.<\/li>\n<li>Simplify if possible.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<ul id=\"eip-809\">\n<li style=\"list-style-type: none\"><\/li>\n<\/ul>\n<ul id=\"eip-578\">\n<li><strong>Solve equations with fraction coefficients by clearing the fractions.<\/strong>\n<ol id=\"eip-id1170323900082\" class=\"stepwise\">\n<li>Find the least common denominator of <em>all<\/em> the fractions in the equation.<\/li>\n<li>Multiply both sides of the equation by that LCD. This clears the fractions.<\/li>\n<li>Solve using the General Strategy for Solving Linear Equations.<\/li>\n<\/ol>\n<\/li>\n<li><strong>Solve an equation with variables and constants on both sides<\/strong>\n<ol id=\"eip-id1170322988380\" class=\"stepwise\">\n<li>Choose one side to be the variable side and then the other will be the constant side.<\/li>\n<li>Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.<\/li>\n<li>Collect the constants to the other side, using the Addition or Subtraction Property of Equality.<\/li>\n<li>Make the coefficient of the variable[latex]1[\/latex], using the Multiplication or Division Property of Equality.<\/li>\n<li>Check the solution by substituting into the original equation.<\/li>\n<\/ol>\n<\/li>\n<li><strong>General strategy for solving linear equations<\/strong>\n<ol id=\"eip-id1170324010914\" class=\"stepwise\">\n<li>Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.<\/li>\n<li>Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.<\/li>\n<li>Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.<\/li>\n<li>Make the coefficient of the variable term equal to [latex]1[\/latex]. Use the Multiplication or Division Property of Equality. State the solution to the equation.<\/li>\n<li>Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\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-3091\">\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><strong>Authored by<\/strong>: Deborah Devlin. <strong>Provided 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><\/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":25777,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"\",\"author\":\"Deborah Devlin\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3091","chapter","type-chapter","status-web-only","hentry"],"part":356,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3091","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3091\/revisions"}],"predecessor-version":[{"id":3845,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3091\/revisions\/3845"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/356"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3091\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=3091"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=3091"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=3091"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=3091"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}