{"id":561,"date":"2023-03-02T19:07:36","date_gmt":"2023-03-02T19:07:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=561"},"modified":"2024-10-18T20:51:28","modified_gmt":"2024-10-18T20:51:28","slug":"geometry-background-youll-need-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/geometry-background-youll-need-1\/","title":{"raw":"Geometry: Background You'll Need 1","rendered":"Geometry: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Solve a simple equation for an unknown&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:7041,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:0,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Solve a simple equation for an unknown<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Equations with One Step<\/h2>\r\n<p>An important property of equations is one that states that you can add the same quantity to both sides of an equation and still maintain an equivalent equation. Sometimes people refer to this as keeping the equation \u201cbalanced.\u201d If you think of an equation as being like a balance scale, the quantities on each side of the equation are equal, or balanced.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>addition property of equality<\/h3>\r\n<p>For all real numbers [latex]a, b[\/latex], and [latex]c[\/latex]: If [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:\r\n\r\n<p><center>[latex]x-6=8[\/latex]<\/center><\/p>\r\n[reveal-answer q=\"746510\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"746510\"]This equation means that if you begin with some unknown number, [latex]x[\/latex], and subtract [latex]6[\/latex], you will end up with [latex]8[\/latex]. You are trying to figure out the value of the variable [latex]x[\/latex].\r\n\r\n\r\n<p>Using the Addition Property of Equality, add [latex]6[\/latex] to both sides of the equation to isolate the variable. You choose to add [latex]6[\/latex] because [latex]6[\/latex] is being subtracted from the variable.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}x-6\\,\\,\\,=\\,\\,\\,\\,8\\\\\\,\\,\\,\\,\\,\\,\\,\\underline{+\\,6\\,\\,\\,\\,\\,\\,\\,\\,+6}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,=\\, 14\\end{array}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:\r\n\r\n<p><center>[latex]x+10=-65[\/latex]<\/center><\/p>\r\n[reveal-answer q=\"846732\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"846732\"]\r\n\r\n\r\n<p style=\"text-align: center;\">[latex]x+10=-65[\/latex]<\/p>\r\n<p>Since [latex]10[\/latex] is being added to the variable, subtract [latex]10[\/latex] from both sides. Note that subtracting [latex]10[\/latex] is the same as adding [latex]\u201310[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}x+10\\,\\,=\\,\\,\\,\\,-65\\\\\\,\\,\\,\\,\\,\\underline{-10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-10}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,=\\,\\,\\,-75\\end{array}[\/latex]<\/p>\r\n<p>To check, substitute the solution, [latex]\u201375[\/latex] for [latex]x[\/latex] in the original equation, then simplify.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,x+10\\,\\,\\,=-65\\\\-75+\\,10\\,\\,\\,=-65\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-65\\,\\,\\,=-65\\end{array}[\/latex]<\/p>\r\n<p>This equation is true, so the solution is correct.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3042[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3043[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Solve a simple equation for an unknown&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:7041,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;11&quot;:0,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Solve a simple equation for an unknown<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Solving Equations with One Step<\/h2>\n<p>An important property of equations is one that states that you can add the same quantity to both sides of an equation and still maintain an equivalent equation. Sometimes people refer to this as keeping the equation \u201cbalanced.\u201d If you think of an equation as being like a balance scale, the quantities on each side of the equation are equal, or balanced.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>addition property of equality<\/h3>\n<p>For all real numbers [latex]a, b[\/latex], and [latex]c[\/latex]: If [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]x-6=8[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q746510\">Show Answer<\/button><\/p>\n<div id=\"q746510\" class=\"hidden-answer\" style=\"display: none\">This equation means that if you begin with some unknown number, [latex]x[\/latex], and subtract [latex]6[\/latex], you will end up with [latex]8[\/latex]. You are trying to figure out the value of the variable [latex]x[\/latex].<\/p>\n<p>Using the Addition Property of Equality, add [latex]6[\/latex] to both sides of the equation to isolate the variable. You choose to add [latex]6[\/latex] because [latex]6[\/latex] is being subtracted from the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}x-6\\,\\,\\,=\\,\\,\\,\\,8\\\\\\,\\,\\,\\,\\,\\,\\,\\underline{+\\,6\\,\\,\\,\\,\\,\\,\\,\\,+6}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,=\\, 14\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]x+10=-65[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q846732\">Show Answer<\/button><\/p>\n<div id=\"q846732\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]x+10=-65[\/latex]<\/p>\n<p>Since [latex]10[\/latex] is being added to the variable, subtract [latex]10[\/latex] from both sides. Note that subtracting [latex]10[\/latex] is the same as adding [latex]\u201310[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}x+10\\,\\,=\\,\\,\\,\\,-65\\\\\\,\\,\\,\\,\\,\\underline{-10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-10}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,=\\,\\,\\,-75\\end{array}[\/latex]<\/p>\n<p>To check, substitute the solution, [latex]\u201375[\/latex] for [latex]x[\/latex] in the original equation, then simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,x+10\\,\\,\\,=-65\\\\-75+\\,10\\,\\,\\,=-65\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-65\\,\\,\\,=-65\\end{array}[\/latex]<\/p>\n<p>This equation is true, so the solution is correct.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3042\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3042&theme=lumen&iframe_resize_id=ohm3042&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3043\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3043&theme=lumen&iframe_resize_id=ohm3043&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Unit 10: Solving Equations and Inequalities, First Edition Developmental Math: An Open Program \",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\" http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/ \",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":71,"module-header":"background_you_need","content_attributions":[{"type":"cc","description":"Unit 10: Solving Equations and Inequalities, First Edition Developmental Math: An Open Program ","author":"","organization":"Monterey Institute of Technology and Education","url":" http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/ ","project":"","license":"cc-by","license_terms":""}],"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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/561"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/16"}],"version-history":[{"count":19,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/561\/revisions"}],"predecessor-version":[{"id":13837,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/561\/revisions\/13837"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/71"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/561\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=561"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=561"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=561"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=561"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}