{"id":2436,"date":"2024-07-25T22:18:02","date_gmt":"2024-07-25T22:18:02","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2436"},"modified":"2025-08-15T16:24:10","modified_gmt":"2025-08-15T16:24:10","slug":"module-14-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-14-background-youll-need-1\/","title":{"raw":"System of Equations With Matrices: Background You'll Need 1","rendered":"System of Equations With Matrices: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use the commutative, associative, and distributive properties of numbers to solve math problems.<\/li>\r\n<\/ul>\r\n<\/section>For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.\r\n<h2>Commutative Properties<\/h2>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.\r\n<div style=\"text-align: center;\">[latex]a+b=b+a[\/latex]<\/div>\r\n<div>\r\n\r\nThe <strong>commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.\r\n<div style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Examples:\r\n<ul>\r\n \t<li>[latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/li>\r\n \t<li>[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/li>\r\n<\/ul>\r\nIt is important to note that neither subtraction nor division is commutative.\r\n\r\nNon-examples:\r\n<ul>\r\n \t<li>[latex]17 - 5[\/latex] is not the same as [latex]5 - 17[\/latex].<\/li>\r\n \t<li>[latex]20\\div 5\\ne 5\\div 20[\/latex].<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24864[\/ohm2_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24865[\/ohm2_question]<\/section>\r\n<h2>Associative Properties<\/h2>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>associative property of multiplication<\/strong> tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.\r\n<div style=\"text-align: center;\">[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/div>\r\n<div>\r\n\r\nThe <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.\r\n<div style=\"text-align: center;\">[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/div>\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Examples:\r\n<ul>\r\n \t<li>[latex]\\left(3\\cdot4\\right)\\cdot5=60\\text{ and }3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/li>\r\n \t<li>[latex][15+\\left(-9\\right)]+23=29\\text{ and }15+[\\left(-9\\right)+23]=29[\/latex]<\/li>\r\n<\/ul>\r\nNon-examples:\r\n<ul>\r\n \t<li>\r\n<div style=\"text-align: center;\">[latex]\\begin{align}8-\\left(3-15\\right) &amp; \\stackrel{?}{=}\\left(8-3\\right)-15 \\\\ 8-\\left(-12\\right) &amp; \\stackrel{?}=5-15 \\\\ 20 &amp; \\neq 20-10 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\n<div style=\"text-align: center;\"><\/div><\/li>\r\n \t<li>\r\n<div style=\"text-align: center;\">[latex]\\begin{align}64\\div\\left(8\\div4\\right)&amp;\\stackrel{?}{=}\\left(64\\div8\\right)\\div4 \\\\ 64\\div2 &amp; \\stackrel{?}{=}8\\div4 \\\\ 32 &amp; \\neq 2 \\\\ \\text{ }\\end{align}[\/latex]<\/div><\/li>\r\n<\/ul>\r\nNote: neither subtraction nor division is associative.\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24866[\/ohm2_question]<\/section>\r\n<h2>Distributive Property<\/h2>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>distributive property<\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.\r\n<div style=\"text-align: center;\">[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\r\n<\/section>This property combines both addition and multiplication (and is the only property to do so).\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Example:\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223815\/CNX_CAT_Figure_01_01_003.jpg\" alt=\"The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.\" width=\"487\" height=\"98\" \/> Example of distributive property[\/caption]\r\n\r\nNon-example:\r\n<p style=\"text-align: center;\">[latex]\\begin{align} 6+\\left(3\\cdot 5\\right)&amp; \\stackrel{?}{=} \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ 6+\\left(15\\right)&amp; \\stackrel{?}{=} \\left(9\\right)\\cdot \\left(11\\right) \\\\ 21&amp; \\ne 99 \\end{align}[\/latex]<\/p>\r\n\r\n<\/section>Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.\r\n\r\nA special case of the distributive property occurs when a sum of terms is subtracted.\r\n<div style=\"text-align: center;\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\r\n<div><section class=\"textbox example\" aria-label=\"Example\">Consider the difference [latex]12-\\left(5+3\\right)[\/latex].\r\n[latex]\\\\[\/latex]\r\nWe can rewrite the difference of the two terms [latex]12[\/latex] and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.\r\n<div style=\"text-align: center;\">[latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\r\nNow, distribute [latex]-1[\/latex] and simplify the result.\r\n<div style=\"text-align: center;\">\r\n<div style=\"text-align: center;\">[latex]\\begin{align}12+\\left(-1\\right)\\cdot\\left(5+3\\right)&amp;=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3] \\\\&amp;=12+(-5-3) \\\\&amp;=12+\\left(-8\\right) \\\\&amp;=4 \\end{align}[\/latex]<\/div>\r\nThis seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}12-\\left(5+3\\right) &amp;=12+\\left(-5-3\\right) \\\\ &amp;=12+\\left(-8\\right) \\\\ &amp;=4\\end{align}[\/latex]<\/div>\r\n<\/div>\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24867[\/ohm2_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24868[\/ohm2_question]<\/section><\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use the commutative, associative, and distributive properties of numbers to solve math problems.<\/li>\n<\/ul>\n<\/section>\n<p>For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.<\/p>\n<h2>Commutative Properties<\/h2>\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.<\/p>\n<div style=\"text-align: center;\">[latex]a+b=b+a[\/latex]<\/div>\n<div>\n<p>The <strong>commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Examples:<\/p>\n<ul>\n<li>[latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/li>\n<li>[latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/li>\n<\/ul>\n<p>It is important to note that neither subtraction nor division is commutative.<\/p>\n<p>Non-examples:<\/p>\n<ul>\n<li>[latex]17 - 5[\/latex] is not the same as [latex]5 - 17[\/latex].<\/li>\n<li>[latex]20\\div 5\\ne 5\\div 20[\/latex].<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24864\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24864&theme=lumen&iframe_resize_id=ohm24864&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24865\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24865&theme=lumen&iframe_resize_id=ohm24865&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Associative Properties<\/h2>\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>associative property of multiplication<\/strong> tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.<\/p>\n<div style=\"text-align: center;\">[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/div>\n<div>\n<p>The <strong>associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.<\/p>\n<div style=\"text-align: center;\">[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Examples:<\/p>\n<ul>\n<li>[latex]\\left(3\\cdot4\\right)\\cdot5=60\\text{ and }3\\cdot\\left(4\\cdot5\\right)=60[\/latex]<\/li>\n<li>[latex][15+\\left(-9\\right)]+23=29\\text{ and }15+[\\left(-9\\right)+23]=29[\/latex]<\/li>\n<\/ul>\n<p>Non-examples:<\/p>\n<ul>\n<li>\n<div style=\"text-align: center;\">[latex]\\begin{align}8-\\left(3-15\\right) & \\stackrel{?}{=}\\left(8-3\\right)-15 \\\\ 8-\\left(-12\\right) & \\stackrel{?}=5-15 \\\\ 20 & \\neq 20-10 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<div style=\"text-align: center;\"><\/div>\n<\/li>\n<li>\n<div style=\"text-align: center;\">[latex]\\begin{align}64\\div\\left(8\\div4\\right)&\\stackrel{?}{=}\\left(64\\div8\\right)\\div4 \\\\ 64\\div2 & \\stackrel{?}{=}8\\div4 \\\\ 32 & \\neq 2 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ul>\n<p>Note: neither subtraction nor division is associative.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24866\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24866&theme=lumen&iframe_resize_id=ohm24866&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Distributive Property<\/h2>\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>distributive property<\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.<\/p>\n<div style=\"text-align: center;\">[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\n<\/section>\n<p>This property combines both addition and multiplication (and is the only property to do so).<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Example:<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223815\/CNX_CAT_Figure_01_01_003.jpg\" alt=\"The number four is separated by a multiplication symbol from a bracketed expression reading: twelve plus negative seven. Arrows extend from the four pointing to the twelve and negative seven separately. This expression equals four times twelve plus four times negative seven. Under this line the expression reads forty eight plus negative twenty eight. Under this line the expression reads twenty as the answer.\" width=\"487\" height=\"98\" \/><figcaption class=\"wp-caption-text\">Example of distributive property<\/figcaption><\/figure>\n<p>Non-example:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} 6+\\left(3\\cdot 5\\right)& \\stackrel{?}{=} \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ 6+\\left(15\\right)& \\stackrel{?}{=} \\left(9\\right)\\cdot \\left(11\\right) \\\\ 21& \\ne 99 \\end{align}[\/latex]<\/p>\n<\/section>\n<p>Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.<\/p>\n<p>A special case of the distributive property occurs when a sum of terms is subtracted.<\/p>\n<div style=\"text-align: center;\">[latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\n<div>\n<section class=\"textbox example\" aria-label=\"Example\">Consider the difference [latex]12-\\left(5+3\\right)[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\nWe can rewrite the difference of the two terms [latex]12[\/latex] and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.<\/p>\n<div style=\"text-align: center;\">[latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\n<p>Now, distribute [latex]-1[\/latex] and simplify the result.<\/p>\n<div style=\"text-align: center;\">\n<div style=\"text-align: center;\">[latex]\\begin{align}12+\\left(-1\\right)\\cdot\\left(5+3\\right)&=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3] \\\\&=12+(-5-3) \\\\&=12+\\left(-8\\right) \\\\&=4 \\end{align}[\/latex]<\/div>\n<p>This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}12-\\left(5+3\\right) &=12+\\left(-5-3\\right) \\\\ &=12+\\left(-8\\right) \\\\ &=4\\end{align}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24867\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24867&theme=lumen&iframe_resize_id=ohm24867&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=\"ohm24868\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24868&theme=lumen&iframe_resize_id=ohm24868&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<\/div>\n","protected":false},"author":12,"menu_order":2,"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\/2436"}],"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":9,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2436\/revisions"}],"predecessor-version":[{"id":7868,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2436\/revisions\/7868"}],"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\/2436\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2436"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2436"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2436"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2436"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}