{"id":16109,"date":"2019-10-01T17:41:06","date_gmt":"2019-10-01T17:41:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-the-distributive-property\/"},"modified":"2020-10-22T09:08:27","modified_gmt":"2020-10-22T09:08:27","slug":"read-the-distributive-property","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-the-distributive-property\/","title":{"raw":"7.1.e - Using the Distributive Property When Solving Equations","rendered":"7.1.e &#8211; Using the Distributive Property When Solving Equations"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the distributive property to solve equations containing parentheses<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 id=\"title2\">The Distributive Property<\/h2>\r\nAs we solve linear equations, we often need to do some work to write\u00a0the linear equations in a form we are familiar with solving.\u00a0This section will focus on manipulating an equation we are asked to solve in such a way that we can use the skills we learned for solving multi-step equations to ultimately arrive at the solution.\r\n\r\nParentheses can\u00a0make solving a problem difficult. To get rid of these unwanted parentheses, we use the distributive property. Using this property, we multiply the number in front of the parentheses by each term inside of the parentheses.\r\n<div class=\"textbox shaded\">\r\n<h3>The Distributive Property of Multiplication<\/h3>\r\nFor all real numbers [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex],\u00a0[latex]a(b+c)=ab+ac[\/latex].\r\n\r\nWhat this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually. Then, you can follow the steps we have already practiced\u00a0to <b>isolate the variable<\/b>\u00a0and solve the equation.\r\n\r\n<\/div>\r\n<h3>Simple distribution and two-step equations<\/h3>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]a[\/latex].\r\n\r\n[latex]4\\left(2a+3\\right)=28[\/latex]\r\n\r\n[reveal-answer q=\"372387\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"372387\"]\r\n\r\nApply the distributive property to expand [latex]4\\left(2a+3\\right)[\/latex] to [latex]8a+12[\/latex]\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4\\left(2a+3\\right)=28\\\\ 8a+12=28\\end{array}[\/latex]<\/p>\r\nSubtract [latex]12[\/latex]\u00a0from both sides to isolate\u00a0the variable term.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}8a+12\\,\\,\\,=\\,\\,\\,28\\\\ \\underline{-12\\,\\,\\,\\,\\,\\,-12}\\\\ 8a\\,\\,\\,=\\,\\,\\,16\\end{array}[\/latex]<\/p>\r\nDivide both terms by [latex]8[\/latex] to get a coefficient of [latex]1[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\underline{8a}=\\underline{16}\\\\8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\\\a\\,=\\,\\,2\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]a=2[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn our next example, we will use the distributive property of multiplication over addition first, simplify, then use the division property to finally solve.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]-3\\left(n - 2\\right)-6=21[\/latex]\r\n\r\nRemember\u2014always simplify each side first.\r\n[reveal-answer q=\"789987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"789987\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468278405\" class=\"unnumbered unstyled\" summary=\"The first line shows negative 3 times parentheses n minus 2 minus 6 equals 21. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-3(n-2)-6=21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]-3n+6-6=21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-3n=21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide both sides by -3 to isolate n.<\/td>\r\n<td>[latex]\\Large\\frac{-3n}{\\color{red}{-3}}\\normalsize =\\Large\\frac{21}{\\color{red}{-3}}[\/latex][latex]n=-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check your answer. Let [latex]n=-7[\/latex] .<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3(n-2)-6=21[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3(\\color{red}{-7}-2)-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3(-9)-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]27-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]21=21\\quad\\checkmark[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a similar problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141911&amp;theme=oea&amp;iframe_resize_id=mom27[\/embed]\r\n\r\n<\/div>\r\nIn the video that follows, we show another example of how to use the distributive property to solve a multi-step linear equation.\r\n\r\nhttps:\/\/youtu.be\/aQOkD8L57V0\r\n<h3>Distribution and combining like terms<\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]3\\left(n - 4\\right)-2n=-3[\/latex]\r\n[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"190834\"]\r\n\r\nSolution:\r\nThe left side of the equation has an expression that we should simplify.\r\n<table id=\"eip-id1168468254328\" class=\"unnumbered unstyled\" summary=\"The top line says \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3(n-4)-2n=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute on the left.<\/td>\r\n<td>[latex]3n-12-2n=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property to rearrange terms.<\/td>\r\n<td>[latex]3n-2n-12=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]n-12=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Isolate <em>n<\/em> using the Addition Property of Equality.<\/td>\r\n<td>[latex]n-12\\color{red}{+12}=-3\\color{red}{+12}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]n=9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.Substitute [latex]n=9[\/latex] into the original equation.\r\n[latex]3(n-4)-2n=-3[\/latex]\r\n[latex]3(\\color{red}{9}-4)-2\\cdot\\color{red}{9}=-3[\/latex]\r\n[latex]3(5)-18=-3[\/latex]\r\n[latex]15-18=-3[\/latex]\r\n[latex]-3=-3\\quad\\checkmark[\/latex]\r\nThe solution checks.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow you can try a few problems that involve distribution.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141737&amp;theme=oea&amp;iframe_resize_id=mom22[\/embed]\r\n\r\n<\/div>\r\n<h3>Distribution and simplifying on both sides<\/h3>\r\nThe next example has expressions on both sides that need to be simplified.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSolve: [latex]2\\left(3k - 1\\right)-5k=-2 - 7[\/latex]\r\n<p class=\"p1\">[reveal-answer q=\"190976\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"190976\"]<\/p>\r\nSolution:\r\nBoth sides of the equation have expressions that we should simplify before we isolate the variable.\r\n<table id=\"eip-id1168469785088\" class=\"unnumbered unstyled\" summary=\"The top line says 2 times parentheses 3k minus 1 minus 5k equals negative 2 minus 7. The next line says \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]2(3k-1)-5k=-2-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute on the left, subtract on the right.<\/td>\r\n<td>[latex]6k-2-5k=-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property of Addition.<\/td>\r\n<td>[latex]6k-5k-2=-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]k-2=-9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Undo subtraction by using the Addition Property of Equality.<\/td>\r\n<td>[latex]k-2\\color{red}{+2}=-9\\color{red}{+2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]k=-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check.Let [latex]k=-7[\/latex].\r\n[latex]2(3k-1)-5k=-2-7[\/latex]\r\n[latex]2(3(\\color{red}{-7}-1)-5(\\color{red}{-7})=-2-7[\/latex]\r\n[latex]2(-21-1)-5(-7)=-9[\/latex]\r\n[latex]2(-22)+35=-9[\/latex]\r\n[latex]-44+35=-9[\/latex]\r\n[latex]-9=-9\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe solution checks.\r\n<p class=\"p1\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nNow, you give it a try!\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141739&amp;theme=oea&amp;iframe_resize_id=mom220[\/embed]\r\n\r\n<\/div>\r\nIn the following video, we present another example of how to solve an equation that requires simplifying before using the addition and subtraction properties.\r\n\r\nhttps:\/\/youtu.be\/shGKzDBA5kQ\r\n\r\n&nbsp;\r\n<h3>Using the distribution property on both sides of the equation<\/h3>\r\nIn the next example, you will see that there are parentheses on both sides of the equal sign, so you will need to use the distributive property twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for [latex]t[\/latex].\r\n\r\n[latex]2\\left(4t-5\\right)=-3\\left(2t+1\\right)[\/latex]\r\n\r\n[reveal-answer q=\"302387\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"302387\"]\r\n\r\nApply the distributive property to expand [latex]2\\left(4t-5\\right)[\/latex] to [latex]8t-10[\/latex] and [latex]-3\\left(2t+1\\right)[\/latex] to [latex]-6t-3[\/latex]. Be careful in this step\u2014you are distributing a negative number, so keep track of the sign of each number after you multiply.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2\\left(4t-5\\right)=-3\\left(2t+1\\right)\\,\\,\\,\\,\\,\\, \\\\ 8t-10=-6t-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd [latex]-6t[\/latex] to both sides to begin combining like terms.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}8t-10=-6t-3\\\\ \\underline{+6t\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+6t}\\,\\,\\,\\,\\,\\,\\,\\\\ 14t-10=\\,\\,\\,\\,-3\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd [latex]10[\/latex] to both sides of the equation to isolate [latex]t[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}14t-10=-3\\\\ \\underline{+10\\,\\,\\,+10}\\\\ 14t=\\,\\,\\,7\\,\\end{array}[\/latex]<\/p>\r\nThe last step is to divide both sides by [latex]14[\/latex] to completely isolate [latex]t[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}14t=7\\,\\,\\,\\,\\\\\\frac{14t}{14}=\\frac{7}{14}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]t=\\frac{1}{2}[\/latex]\r\n\r\nWe simplified the fraction [latex]\\frac{7}{14}[\/latex] into [latex]\\frac{1}{2}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]39431[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video, we solve another multi-step equation with two sets of parentheses.\r\n\r\nhttps:\/\/youtu.be\/StomYTb7Xb8\r\n\r\n&nbsp;","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the distributive property to solve equations containing parentheses<\/li>\n<\/ul>\n<\/div>\n<h2 id=\"title2\">The Distributive Property<\/h2>\n<p>As we solve linear equations, we often need to do some work to write\u00a0the linear equations in a form we are familiar with solving.\u00a0This section will focus on manipulating an equation we are asked to solve in such a way that we can use the skills we learned for solving multi-step equations to ultimately arrive at the solution.<\/p>\n<p>Parentheses can\u00a0make solving a problem difficult. To get rid of these unwanted parentheses, we use the distributive property. Using this property, we multiply the number in front of the parentheses by each term inside of the parentheses.<\/p>\n<div class=\"textbox shaded\">\n<h3>The Distributive Property of Multiplication<\/h3>\n<p>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>What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually. Then, you can follow the steps we have already practiced\u00a0to <b>isolate the variable<\/b>\u00a0and solve the equation.<\/p>\n<\/div>\n<h3>Simple distribution and two-step equations<\/h3>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]a[\/latex].<\/p>\n<p>[latex]4\\left(2a+3\\right)=28[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q372387\">Show Solution<\/span><\/p>\n<div id=\"q372387\" class=\"hidden-answer\" style=\"display: none\">\n<p>Apply the distributive property to expand [latex]4\\left(2a+3\\right)[\/latex] to [latex]8a+12[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4\\left(2a+3\\right)=28\\\\ 8a+12=28\\end{array}[\/latex]<\/p>\n<p>Subtract [latex]12[\/latex]\u00a0from both sides to isolate\u00a0the variable term.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}8a+12\\,\\,\\,=\\,\\,\\,28\\\\ \\underline{-12\\,\\,\\,\\,\\,\\,-12}\\\\ 8a\\,\\,\\,=\\,\\,\\,16\\end{array}[\/latex]<\/p>\n<p>Divide both terms by [latex]8[\/latex] to get a coefficient of [latex]1[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}\\underline{8a}=\\underline{16}\\\\8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\\\a\\,=\\,\\,2\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]a=2[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<p>In our next example, we will use the distributive property of multiplication over addition first, simplify, then use the division property to finally solve.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]-3\\left(n - 2\\right)-6=21[\/latex]<\/p>\n<p>Remember\u2014always simplify each side first.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q789987\">Show Solution<\/span><\/p>\n<div id=\"q789987\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468278405\" class=\"unnumbered unstyled\" summary=\"The first line shows negative 3 times parentheses n minus 2 minus 6 equals 21. The next line says\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-3(n-2)-6=21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]-3n+6-6=21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-3n=21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide both sides by -3 to isolate n.<\/td>\n<td>[latex]\\Large\\frac{-3n}{\\color{red}{-3}}\\normalsize =\\Large\\frac{21}{\\color{red}{-3}}[\/latex][latex]n=-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check your answer. Let [latex]n=-7[\/latex] .<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]-3(n-2)-6=21[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]-3(\\color{red}{-7}-2)-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]-3(-9)-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]27-6\\stackrel{\\text{?}}{=}21[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]21=21\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a similar problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141911\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141911&#38;theme=oea&#38;iframe_resize_id=ohm141911&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the video that follows, we show another example of how to use the distributive property to solve a multi-step linear equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solving an Equation with One Set of Parentheses\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/aQOkD8L57V0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>Distribution and combining like terms<\/h3>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]3\\left(n - 4\\right)-2n=-3[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q190834\">Show Solution<\/span><\/p>\n<div id=\"q190834\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nThe left side of the equation has an expression that we should simplify.<\/p>\n<table id=\"eip-id1168468254328\" class=\"unnumbered unstyled\" summary=\"The top line says\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3(n-4)-2n=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute on the left.<\/td>\n<td>[latex]3n-12-2n=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property to rearrange terms.<\/td>\n<td>[latex]3n-2n-12=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]n-12=-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Isolate <em>n<\/em> using the Addition Property of Equality.<\/td>\n<td>[latex]n-12\\color{red}{+12}=-3\\color{red}{+12}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]n=9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.Substitute [latex]n=9[\/latex] into the original equation.<br \/>\n[latex]3(n-4)-2n=-3[\/latex]<br \/>\n[latex]3(\\color{red}{9}-4)-2\\cdot\\color{red}{9}=-3[\/latex]<br \/>\n[latex]3(5)-18=-3[\/latex]<br \/>\n[latex]15-18=-3[\/latex]<br \/>\n[latex]-3=-3\\quad\\checkmark[\/latex]<br \/>\nThe solution checks.<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Now you can try a few problems that involve distribution.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141737\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141737&#38;theme=oea&#38;iframe_resize_id=ohm141737&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3>Distribution and simplifying on both sides<\/h3>\n<p>The next example has expressions on both sides that need to be simplified.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Solve: [latex]2\\left(3k - 1\\right)-5k=-2 - 7[\/latex]<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q190976\">Show Solution<\/span><\/p>\n<p class=\"p1\">\n<div id=\"q190976\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nBoth sides of the equation have expressions that we should simplify before we isolate the variable.<\/p>\n<table id=\"eip-id1168469785088\" class=\"unnumbered unstyled\" summary=\"The top line says 2 times parentheses 3k minus 1 minus 5k equals negative 2 minus 7. The next line says\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]2(3k-1)-5k=-2-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute on the left, subtract on the right.<\/td>\n<td>[latex]6k-2-5k=-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property of Addition.<\/td>\n<td>[latex]6k-5k-2=-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]k-2=-9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Undo subtraction by using the Addition Property of Equality.<\/td>\n<td>[latex]k-2\\color{red}{+2}=-9\\color{red}{+2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]k=-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check.Let [latex]k=-7[\/latex].<br \/>\n[latex]2(3k-1)-5k=-2-7[\/latex]<br \/>\n[latex]2(3(\\color{red}{-7}-1)-5(\\color{red}{-7})=-2-7[\/latex]<br \/>\n[latex]2(-21-1)-5(-7)=-9[\/latex]<br \/>\n[latex]2(-22)+35=-9[\/latex]<br \/>\n[latex]-44+35=-9[\/latex]<br \/>\n[latex]-9=-9\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The solution checks.<\/p>\n<p class=\"p1\"><\/div>\n<\/div>\n<\/div>\n<p>Now, you give it a try!<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141739\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141739&#38;theme=oea&#38;iframe_resize_id=ohm141739&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video, we present another example of how to solve an equation that requires simplifying before using the addition and subtraction properties.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Solve Linear Equations in One Variable with Simplifying (One-Step Add\/Subtract)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/shGKzDBA5kQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h3>Using the distribution property on both sides of the equation<\/h3>\n<p>In the next example, you will see that there are parentheses on both sides of the equal sign, so you will need to use the distributive property twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for [latex]t[\/latex].<\/p>\n<p>[latex]2\\left(4t-5\\right)=-3\\left(2t+1\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q302387\">Show Solution<\/span><\/p>\n<div id=\"q302387\" class=\"hidden-answer\" style=\"display: none\">\n<p>Apply the distributive property to expand [latex]2\\left(4t-5\\right)[\/latex] to [latex]8t-10[\/latex] and [latex]-3\\left(2t+1\\right)[\/latex] to [latex]-6t-3[\/latex]. Be careful in this step\u2014you are distributing a negative number, so keep track of the sign of each number after you multiply.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}2\\left(4t-5\\right)=-3\\left(2t+1\\right)\\,\\,\\,\\,\\,\\, \\\\ 8t-10=-6t-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Add [latex]-6t[\/latex] to both sides to begin combining like terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}8t-10=-6t-3\\\\ \\underline{+6t\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+6t}\\,\\,\\,\\,\\,\\,\\,\\\\ 14t-10=\\,\\,\\,\\,-3\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Add [latex]10[\/latex] to both sides of the equation to isolate [latex]t[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}14t-10=-3\\\\ \\underline{+10\\,\\,\\,+10}\\\\ 14t=\\,\\,\\,7\\,\\end{array}[\/latex]<\/p>\n<p>The last step is to divide both sides by [latex]14[\/latex] to completely isolate [latex]t[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}14t=7\\,\\,\\,\\,\\\\\\frac{14t}{14}=\\frac{7}{14}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]t=\\frac{1}{2}[\/latex]<\/p>\n<p>We simplified the fraction [latex]\\frac{7}{14}[\/latex] into [latex]\\frac{1}{2}[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm39431\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=39431&theme=oea&iframe_resize_id=ohm39431&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video, we solve another multi-step equation with two sets of parentheses.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Solving an Equation with Parentheses on Both Sides\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/StomYTb7Xb8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\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-16109\">\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>Revision and Adaptation. <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><li>Solving an Equation with One Set of Parentheses. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/aQOkD8L57V0\">https:\/\/youtu.be\/aQOkD8L57V0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solving an Equation with Parentheses on Both Sides. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/StomYTb7Xb8\">https:\/\/youtu.be\/StomYTb7Xb8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solving an Equation with Fractions (Clear Fractions). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/AvJTPeACTY0\">https:\/\/youtu.be\/AvJTPeACTY0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solving an Equation with Decimals (Clear Decimals). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/wtwepTZZnlY\">https:\/\/youtu.be\/wtwepTZZnlY<\/a>. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <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":169554,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 10: Solving Equations and Inequalities, from 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\":\"\"},{\"type\":\"original\",\"description\":\"Solving an Equation with One Set of 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