{"id":2438,"date":"2024-07-25T22:18:54","date_gmt":"2024-07-25T22:18:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2438"},"modified":"2024-11-21T22:25:41","modified_gmt":"2024-11-21T22:25:41","slug":"module-14-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-14-background-youll-need-2\/","title":{"raw":"System of Equations With Matrices: Background You'll Need 2","rendered":"System of Equations With Matrices: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use the identity and inverse properties of numbers to solve math problems.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Identity Properties<\/h2>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>identity property of addition<\/strong> states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.\r\n<div style=\"text-align: center;\">\u00a0[latex]a+0=a[\/latex]<\/div>\r\n<div><\/div>\r\nThe <strong>identity property of multiplication<\/strong> states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.\r\n<div style=\"text-align: center;\">[latex]a\\cdot 1=a[\/latex]<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<ul>\r\n \t<li>[latex]\\left(-6\\right)+0=-6[\/latex]<\/li>\r\n \t<li>[latex]23\\cdot 1=23[\/latex]<\/li>\r\n<\/ul>\r\nNote: There are no exceptions for these properties; they work for every real number, including [latex]0[\/latex] and [latex]1[\/latex].\r\n\r\n<\/section>\r\n<h2>Inverse Properties<\/h2>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>inverse property of addition<\/strong> states that, for every real number [latex]a[\/latex], there is a unique number, called the additive inverse (or opposite), denoted [latex]\u2212a[\/latex], that, when added to the original number, results in the additive identity, [latex]0[\/latex].\r\n<div style=\"text-align: center;\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\r\n<div>\r\n\r\nThe <strong>inverse property of multiplication<\/strong> holds for all real numbers except [latex]0[\/latex] because the reciprocal of [latex]0[\/latex] is not defined. The property states that, for every real number [latex]a[\/latex], there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\frac{1}{a}[\/latex], that, when multiplied by the original number, results in the multiplicative identity, [latex]1[\/latex].\r\n<div style=\"text-align: center;\">\r\n<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=1[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<ul>\r\n \t<li>If [latex]a=-8[\/latex], the additive inverse is [latex]8[\/latex], since [latex]\\left(-8\\right)+8=0[\/latex].<\/li>\r\n \t<li>If [latex]a=-\\frac{2}{3}[\/latex], the reciprocal, denoted [latex]\\frac{1}{a}[\/latex], is [latex]-\\frac{3}{2}[\/latex] because\r\n<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=\\left(-\\dfrac{2}{3}\\right)\\cdot \\left(-\\dfrac{3}{2}\\right)=1[\/latex]<\/div><\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24869[\/ohm2_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24870[\/ohm2_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24871[\/ohm2_question]<\/section><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use the identity and inverse properties of numbers to solve math problems.<\/li>\n<\/ul>\n<\/section>\n<h2>Identity Properties<\/h2>\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>identity property of addition<\/strong> states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.<\/p>\n<div style=\"text-align: center;\">\u00a0[latex]a+0=a[\/latex]<\/div>\n<div><\/div>\n<p>The <strong>identity property of multiplication<\/strong> states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot 1=a[\/latex]<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<ul>\n<li>[latex]\\left(-6\\right)+0=-6[\/latex]<\/li>\n<li>[latex]23\\cdot 1=23[\/latex]<\/li>\n<\/ul>\n<p>Note: There are no exceptions for these properties; they work for every real number, including [latex]0[\/latex] and [latex]1[\/latex].<\/p>\n<\/section>\n<h2>Inverse Properties<\/h2>\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>inverse property of addition<\/strong> states that, for every real number [latex]a[\/latex], there is a unique number, called the additive inverse (or opposite), denoted [latex]\u2212a[\/latex], that, when added to the original number, results in the additive identity, [latex]0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\n<div>\n<p>The <strong>inverse property of multiplication<\/strong> holds for all real numbers except [latex]0[\/latex] because the reciprocal of [latex]0[\/latex] is not defined. The property states that, for every real number [latex]a[\/latex], there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\frac{1}{a}[\/latex], that, when multiplied by the original number, results in the multiplicative identity, [latex]1[\/latex].<\/p>\n<div style=\"text-align: center;\">\n<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<ul>\n<li>If [latex]a=-8[\/latex], the additive inverse is [latex]8[\/latex], since [latex]\\left(-8\\right)+8=0[\/latex].<\/li>\n<li>If [latex]a=-\\frac{2}{3}[\/latex], the reciprocal, denoted [latex]\\frac{1}{a}[\/latex], is [latex]-\\frac{3}{2}[\/latex] because\n<div style=\"text-align: center;\">[latex]a\\cdot \\dfrac{1}{a}=\\left(-\\dfrac{2}{3}\\right)\\cdot \\left(-\\dfrac{3}{2}\\right)=1[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24869\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24869&theme=lumen&iframe_resize_id=ohm24869&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24870\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24870&theme=lumen&iframe_resize_id=ohm24870&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24871\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24871&theme=lumen&iframe_resize_id=ohm24871&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n","protected":false},"author":12,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":327,"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\/2438"}],"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":5,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2438\/revisions"}],"predecessor-version":[{"id":4435,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2438\/revisions\/4435"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/327"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2438\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2438"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2438"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2438"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2438"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}