{"id":9023,"date":"2017-05-01T15:08:24","date_gmt":"2017-05-01T15:08:24","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9023"},"modified":"2020-10-22T09:09:26","modified_gmt":"2020-10-22T09:09:26","slug":"summary-solve-equations-using-the-division-and-multiplication-properties-of-equality","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/summary-solve-equations-using-the-division-and-multiplication-properties-of-equality\/","title":{"raw":"7.1.i - Summary: Solving Equations Using the Properties of Equality","rendered":"7.1.i &#8211; Summary: Solving Equations Using the Properties of Equality"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<strong>Determine whether a number is a solution to an equation.<\/strong>\r\n<ol id=\"eip-id1170325336254\" 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.<\/li>\r\n<\/ol>\r\n<p style=\"padding-left: 60px\">If it is true, the number is a solution.\r\nIf it is not true, the number is not a solution.<\/p>\r\n<strong>Subtraction and Addition Properties of Equality<\/strong>\r\n<ul id=\"eip-id1170322764024\">\r\n \t<li><strong>Subtraction Property of Equality<\/strong><\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 90px\">For all real numbers <em> a, b,<\/em> and <em>c<\/em>,\r\nif <em> a = b<\/em> then [latex]a-c=b-c[\/latex] .<\/p>\r\n\r\n<ul id=\"eip-id1170322764024\">\r\n \t<li><strong>Addition Property of Equality<\/strong><\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 90px\">For all real numbers <em> a, b,<\/em> and <em>c<\/em>,\r\nif <em> a = b<\/em> then [latex]a+c=b+c[\/latex] .<\/p>\r\n<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<strong>Problem-solving strategy<\/strong>\r\n<ol id=\"eip-id1170325411872\" class=\"stepwise\">\r\n \t<li>Read the problem. Make sure you understand all the words and ideas.<\/li>\r\n \t<li>Identify what you are looking for.<\/li>\r\n \t<li>Name what you are looking for. Choose a variable to represent that quantity.<\/li>\r\n \t<li>Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.<\/li>\r\n \t<li>Solve the equation using good algebra techniques.<\/li>\r\n \t<li>Check the answer in the problem and make sure it makes sense.<\/li>\r\n \t<li>Answer the question with a complete sentence.<\/li>\r\n<\/ol>\r\n<strong>Division and Multiplication Properties of Equality<\/strong>\r\n<ul id=\"eip-id1170323851320\">\r\n \t<li><strong>Division Property of Equality:<\/strong>\r\n<ul>\r\n \t<li>For all real numbers <em>a, b, c,<\/em> and [latex]c\\ne 0[\/latex] , if [latex]a=b[\/latex] , then [latex]ac=bc[\/latex] .<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Multiplication Property of Equality:<\/strong>\r\n<ul>\r\n \t<li>For all real numbers <em>a, b, c,<\/em> if [latex]a=b[\/latex] , then [latex]ac=bc[\/latex] .<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<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<strong>The Distributive Property of Multiplication<\/strong>\r\n<p style=\"padding-left: 30px\">For all real numbers [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex],\u00a0[latex]a(b+c)=ab+ac[\/latex].<\/p>\r\n<strong>Solve equations by clearing the Denominators<\/strong>\r\n<ol id=\"eip-id1168466140919\" 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>Isolate the variable terms on one side, and the constant terms on the other side.<\/li>\r\n \t<li>Simplify both sides.<\/li>\r\n \t<li>Use the multiplication or division property to make the coefficient on the variable equal to [latex]1[\/latex].<\/li>\r\n<\/ol>\r\n<strong>Solving Equations of the Form [latex]|x|=a[\/latex]<\/strong>\r\n<p style=\"padding-left: 30px\"><span class=\"tight\">For any positive number [latex]a[\/latex], the solution of [latex]\\left|x\\right|=a[\/latex]\u00a0is [latex]x=a[\/latex]\u00a0 or\u00a0 [latex]x=\u2212a[\/latex].\u00a0 [latex]x[\/latex] can be a single variable or any algebraic expression.<\/span><\/p>\r\n<strong>General strategy for solving linear equations<\/strong>\r\n<ol>\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<strong>Solutions to equations can fall into three categories:<\/strong>\r\n<ol>\r\n \t<li>One solution. This is when you find the only value of the variable, such as [latex]x = 5[\/latex].<\/li>\r\n \t<li>No solution, DNE (does not exist). This is when a false statement appears, like [latex]4 = 7[\/latex].<\/li>\r\n \t<li>Many solutions, also called infinitely many solutions or All Real Numbers. This is when a true statement appears, like [latex]x + 3 = x + 3[\/latex].<\/li>\r\n<\/ol>\r\n<ol class=\"stepwise\">\r\n \t<li style=\"list-style-type: none\"><\/li>\r\n<\/ol>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170326208984\" class=\"definition\">\r\n \t<dt><strong>solution of an equation<\/strong><\/dt>\r\n \t<dd id=\"fs-id1170323908665\">A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.<\/dd>\r\n \t<dt><strong>isolate a variable<\/strong><\/dt>\r\n \t<dd id=\"fs-id1170323908665\">To isolate a variable means to rewrite an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.<\/dd>\r\n<\/dl>","rendered":"<h2>Key Concepts<\/h2>\n<p><strong>Determine whether a number is a solution to an equation.<\/strong><\/p>\n<ol id=\"eip-id1170325336254\" 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.<\/li>\n<\/ol>\n<p style=\"padding-left: 60px\">If it is true, the number is a solution.<br \/>\nIf it is not true, the number is not a solution.<\/p>\n<p><strong>Subtraction and Addition Properties of Equality<\/strong><\/p>\n<ul id=\"eip-id1170322764024\">\n<li><strong>Subtraction Property of Equality<\/strong><\/li>\n<\/ul>\n<p style=\"padding-left: 90px\">For all real numbers <em> a, b,<\/em> and <em>c<\/em>,<br \/>\nif <em> a = b<\/em> then [latex]a-c=b-c[\/latex] .<\/p>\n<ul id=\"eip-id1170322764024\">\n<li><strong>Addition Property of Equality<\/strong><\/li>\n<\/ul>\n<p style=\"padding-left: 90px\">For all real numbers <em> a, b,<\/em> and <em>c<\/em>,<br \/>\nif <em> a = b<\/em> then [latex]a+c=b+c[\/latex] .<\/p>\n<p><strong>Translate a word sentence to an algebraic equation.<\/strong><\/p>\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<p><strong>Problem-solving strategy<\/strong><\/p>\n<ol id=\"eip-id1170325411872\" class=\"stepwise\">\n<li>Read the problem. Make sure you understand all the words and ideas.<\/li>\n<li>Identify what you are looking for.<\/li>\n<li>Name what you are looking for. Choose a variable to represent that quantity.<\/li>\n<li>Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.<\/li>\n<li>Solve the equation using good algebra techniques.<\/li>\n<li>Check the answer in the problem and make sure it makes sense.<\/li>\n<li>Answer the question with a complete sentence.<\/li>\n<\/ol>\n<p><strong>Division and Multiplication Properties of Equality<\/strong><\/p>\n<ul id=\"eip-id1170323851320\">\n<li><strong>Division Property of Equality:<\/strong>\n<ul>\n<li>For all real numbers <em>a, b, c,<\/em> and [latex]c\\ne 0[\/latex] , if [latex]a=b[\/latex] , then [latex]ac=bc[\/latex] .<\/li>\n<\/ul>\n<\/li>\n<li><strong>Multiplication Property of Equality:<\/strong>\n<ul>\n<li>For all real numbers <em>a, b, c,<\/em> if [latex]a=b[\/latex] , then [latex]ac=bc[\/latex] .<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>Solve an equation with variables and constants on both sides<\/strong><\/p>\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<p><strong>The Distributive Property of Multiplication<\/strong><\/p>\n<p style=\"padding-left: 30px\">For all real numbers [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex],\u00a0[latex]a(b+c)=ab+ac[\/latex].<\/p>\n<p><strong>Solve equations by clearing the Denominators<\/strong><\/p>\n<ol id=\"eip-id1168466140919\" 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>Isolate the variable terms on one side, and the constant terms on the other side.<\/li>\n<li>Simplify both sides.<\/li>\n<li>Use the multiplication or division property to make the coefficient on the variable equal to [latex]1[\/latex].<\/li>\n<\/ol>\n<p><strong>Solving Equations of the Form [latex]|x|=a[\/latex]<\/strong><\/p>\n<p style=\"padding-left: 30px\"><span class=\"tight\">For any positive number [latex]a[\/latex], the solution of [latex]\\left|x\\right|=a[\/latex]\u00a0is [latex]x=a[\/latex]\u00a0 or\u00a0 [latex]x=\u2212a[\/latex].\u00a0 [latex]x[\/latex] can be a single variable or any algebraic expression.<\/span><\/p>\n<p><strong>General strategy for solving linear equations<\/strong><\/p>\n<ol>\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<p><strong>Solutions to equations can fall into three categories:<\/strong><\/p>\n<ol>\n<li>One solution. This is when you find the only value of the variable, such as [latex]x = 5[\/latex].<\/li>\n<li>No solution, DNE (does not exist). This is when a false statement appears, like [latex]4 = 7[\/latex].<\/li>\n<li>Many solutions, also called infinitely many solutions or All Real Numbers. This is when a true statement appears, like [latex]x + 3 = x + 3[\/latex].<\/li>\n<\/ol>\n<ol class=\"stepwise\">\n<li style=\"list-style-type: none\"><\/li>\n<\/ol>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170326208984\" class=\"definition\">\n<dt><strong>solution of an equation<\/strong><\/dt>\n<dd id=\"fs-id1170323908665\">A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.<\/dd>\n<dt><strong>isolate a variable<\/strong><\/dt>\n<dd id=\"fs-id1170323908665\">To isolate a variable means to rewrite an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.<\/dd>\n<\/dl>\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-9023\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><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":11,"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\"}]","CANDELA_OUTCOMES_GUID":"0bdb5d89bfce4d0ebafbee429e80baf4","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-9023","chapter","type-chapter","status-publish","hentry"],"part":7476,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9023","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":20,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9023\/revisions"}],"predecessor-version":[{"id":20304,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9023\/revisions\/20304"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/7476"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/9023\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/media?parent=9023"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=9023"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=9023"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/license?post=9023"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}