{"id":6590,"date":"2020-10-01T16:29:52","date_gmt":"2020-10-01T16:29:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6590"},"modified":"2021-06-01T22:58:28","modified_gmt":"2021-06-01T22:58:28","slug":"1-6-solving-equations-containing-decimals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/1-6-solving-equations-containing-decimals\/","title":{"raw":"1.6 Solving Equations Containing Decimals","rendered":"1.6 Solving Equations Containing Decimals"},"content":{"raw":"
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section 1.6 Learning Objectives<\/h3>\r\n1.6: Solving Equations Containing Decimals<\/strong>\r\n
    \r\n \t
  • Solve one-step equations containing decimals<\/li>\r\n \t
  • Solve multi-step equations containing decimals<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n
    \r\n

    think about it<\/h3>\r\nCan you determine\u00a0what you would do differently if you were asked to solve equations like these?\r\n\r\nSolve [latex]{12.5}+{ t }= {-7.5}[\/latex].\r\n\r\nWhat makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would solve this equation with decimals.\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n\r\n[reveal-answer q=\"680980\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"680980\"]To solve this equation you need to remember how to\u00a0add or subtract decimal numbers. You also need to remember that when you subtract a number from a negative number, your result will be negative.\r\n\r\nUsing the Addition Property of Equality, subtract 12.5 from both sides of the equation to isolate the variable, t<\/em>. You choose to\u00a0subtract\u00a012.5 because\u00a012.5 is a positive addend.\r\n

    [latex] \\displaystyle \\begin{array}{r}{12.5}+{t}\\,\\,\\,=\\,\\,\\,\\,{-7.5}\\\\\\,\\,\\,\\,\\,\\,\\,\\underline{-12.5\\,\\,\\,\\,\\,\\,\\,\\,-12.5}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,t\\,\\,=\\, -20\\end{array}[\/latex]<\/p>\r\n

    To add two numbers of the same sign,\u00a0first add their absolute values:<\/p>\r\n

    [latex]\\begin{array}{r}\\left|-12.5\\right| = 12.5\\\\\\left|-7.5\\right| = 7.5\\,\\,\\,\\\\12.5 + 7.5 = 20\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n

    Now apply the sign they share, which is negative:<\/p>\r\n

    [latex]-12.5 -7.5 = -20[\/latex]<\/p>\r\n

    [\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n \r\n

    Solve one-step equations containing decimals<\/h2>\r\nIn the following video, two examples of using the addition property of equality with decimal numbers are shown.\r\n\r\nhttps:\/\/youtu.be\/D8wKGlxf6bM\r\n

    Solve multi-step equations containing decimals<\/h2>\r\nSometimes, you will encounter a multi-step <\/strong>equation with decimals. If you prefer not working with decimals, you can use the multiplication property of equality to multiply both sides of the equation by a a factor of 10 that will help clear the decimals. See the example below.\r\n
    \r\n

    Example 1<\/h3>\r\nSolve [latex]3y+10.5=6.5+2.5y[\/latex] by clearing the decimals in the equation first.\r\n\r\n[reveal-answer q=\"159951\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"159951\"]\r\n\r\nSince moving each decimal one time to the right for our decimal terms would clear them, we want to multiply every term\u00a0<\/strong>by 10 to clear\u00a0the decimals from the equation. (Remember, we must multiply EVERY term by 10, even if it is not a decimal).\r\n

    [latex]\\begin{array}{r}3y+10.5=6.5+2.5y\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\\\ 10\\left(3y+10.5\\right)=10\\left(6.5+2.5y\\right)\\end{array}[\/latex]<\/p>\r\nUse the distributive property to expand the expressions on both sides.\r\n

    [latex]\\begin{array}{r}10\\left(3y\\right)+10\\left(10.5\\right)=10\\left(6.5\\right)+10\\left(2.5y\\right)\\end{array}[\/latex]<\/p>\r\nMultiply.\r\n

    [latex]30y+105=65+25y[\/latex]<\/p>\r\nMove the smaller variable term, [latex]25y[\/latex], by subtracting it from both sides.\r\n

    [latex]\\begin{array}{r}30y+105=65+25y\\,\\,\\\\ \\underline{-25y\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-25y} \\\\5y+105=65\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nSubtract 105 from both sides to isolate the term with the variable.\r\n

    [latex]\\begin{array}{r}5y+105=65\\,\\,\\,\\\\ \\underline{\\,\\,\\,\\,\\,\\,-105\\,-105} \\\\5y=-40\\end{array}[\/latex]<\/p>\r\nDivide both sides by 5 to isolate the y<\/em>.\r\n

    [latex]\\begin{array}{l}\\underline{5y}=\\underline{-40}\\\\ 5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,5\\\\ \\,\\,\\,y=-8\\end{array}[\/latex]<\/p>\r\n\r\n

    Answer<\/h4>\r\n[latex]y=-8[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following example, you will see the same problem as the last example, but solved\u00a0without\u00a0<\/em>clearing the decimals. It might be helpful to compare the two methods and decide which method you prefer.\r\n
    \r\n

    Example 2<\/h3>\r\nSolve [latex]3y+10.5=6.5+2.5y[\/latex]\u00a0without\u00a0<\/em>clearing decimals\r\n\r\n[reveal-answer q=\"581911\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"581911\"]\r\n

    [latex]3y+10.5=6.5+2.5y[\/latex]<\/p>\r\nSubtract [latex]2.5y[\/latex] from both sides.\r\n

    [latex]\\begin{array}{r}3y+10.5=6.5+2.5y\\,\\,\\\\ \\underline{-2.5y\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2.5y} \\\\0.5y+10.5=6.5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nSubtract 10.5 from both sides to isolate the term with the variable.\r\n

    [latex]\\begin{array}{r}0.5y+10.5=6.5\\,\\,\\,\\\\ \\underline{\\,\\,\\,\\,\\,\\,-10.5\\,-10.5} \\\\0.5y=-4\\end{array}[\/latex]<\/p>\r\nDivide both sides by 0.5 to isolate the y<\/em>.\r\n

    [latex]\\begin{array}{l}\\underline{0.5y}=\\underline{-4}\\\\ 0.5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0.5\\\\ \\,\\,\\,y=-8\\end{array}[\/latex]<\/p>\r\n\r\n

    Answer<\/h4>\r\n[latex]y=-8[\/latex][\/hidden-answer]\r\n\r\n \r\n\r\n<\/div>\r\n \r\n\r\nIn the following video, we show another example of clearing decimals first to solve a multi-step linear equation.\r\n\r\nhttps:\/\/youtu.be\/wtwepTZZnlY\r\n\r\nHere is a reminder of the steps to follow when you solve multi-step equations.\r\n
    \r\n

    Solving Multi-Step Equations<\/h3>\r\n1. Simplify each side by clearing parentheses and combining like terms.\r\n\r\n2. (Optional) Multiply to clear any fractions or decimals.\r\n\r\n3. Add or subtract to isolate the variable term\u2014you may have to move a term with the variable.\r\n\r\n4. Multiply or divide to isolate the variable.\r\n\r\n5. Check the solution.\r\n\r\n<\/div>\r\nWe conclude with an example that requires step 1 - clearing parentheses through distribution.\r\n
    \r\n

    Example 3<\/h3>\r\nSolve:\u00a0 [latex]0.7(x+1.02)+0.2x=0.5(x-1.2)-6.24[\/latex]\r\n\r\n[reveal-answer q=\"116238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"116238\"]\r\n\r\nWe will look at two solutions to this problem, first without clearing the decimals and the second with clearing decimals.\r\n\r\nSolution 1 (Without Clearing Decimals):\u00a0\u00a0<\/strong>\r\n\r\nBegin by distributing and proceed by following our standard solving steps.\r\n

    [latex]0.7(x+1.02)+0.2x=0.5(x-1.2)-6.24[\/latex]<\/p>\r\n

    [latex]0.7x+0.714+0.2x=0.5x-0.6-6.24[\/latex]<\/p>\r\n

    [latex]0.9x+0.714=0.5x-6.84[\/latex]<\/p>\r\n

    [latex]\\underline{\\hspace{-.05in} -0.5x\\hspace{.65in}-0.5x\\hspace{.63in}}[\/latex]<\/p>\r\n

    [latex]0.4x+0.714=-6.84\\hspace{.45in}[\/latex]<\/p>\r\n

    [latex]\\underline{\\hspace{.4in} -0.714 \\hspace{.2in} -0.714 \\hspace{.1in}}\\hspace{.25in}[\/latex]<\/p>\r\n

    [latex]0.4x\\hspace{.62in}=-7.554\\hspace{.34in}[\/latex]<\/p>\r\n

    [latex]\\frac{0.4x}{0.4}=\\frac{-7.554}{0.4}[\/latex]<\/p>\r\n

    [latex]x=-18.885[\/latex]<\/p>\r\nSolution 2 (With Clearing Decimals)<\/strong>:\u00a0 This time we will clear the decimals. However, we still want to distribute first to remove the parentheses.\r\n

    [latex]0.7(x+1.02)+0.2x=0.5(x-1.2)-6.24[\/latex]<\/p>\r\n

    [latex]0.7x+0.714+0.2x=0.5x-0.6-6.24[\/latex]<\/p>\r\nWe will multiply each term by 1000 to clear the decimals. We need to multiply by 1000 because the term with the most decimal places has three decimal places.\r\n

    [latex]1000(0.7x)+1000(0.714)+1000(0.2x)=1000(0.5x)-1000(0.6)-1000(6.24)[\/latex]<\/p>\r\n

    [latex]700x+714+200x=500x-600-6240[\/latex]<\/p>\r\n

    [latex]900x+714=500x-6840[\/latex]<\/p>\r\n

    [latex]\\underline{-500x \\hspace{.5in} -500x \\hspace{.56in}}\\hspace{.1in}[\/latex]<\/p>\r\n

    [latex]400x+714=-6840\\hspace{.4in}[\/latex]<\/p>\r\n

    [latex]\\underline{\\hspace{.47in}-714\\hspace{.2in}-714}\\hspace{.47in}[\/latex]<\/p>\r\n

    [latex]400x\\hspace{.47in} =-7554 \\hspace{.38in}[\/latex]<\/p>\r\n

    [latex]\\frac{400x}{400}=\\frac{-7554}{400}[\/latex]<\/p>\r\n

    [latex]x=-18.885[\/latex]<\/p>\r\n\r\n

    Answer<\/strong><\/span><\/h4>\r\n[latex]x=-18.885[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n ","rendered":"
    \n

    section 1.6 Learning Objectives<\/h3>\n

    1.6: Solving Equations Containing Decimals<\/strong><\/p>\n

      \n
    • Solve one-step equations containing decimals<\/li>\n
    • Solve multi-step equations containing decimals<\/li>\n<\/ul>\n<\/div>\n

       <\/p>\n

      \n

      think about it<\/h3>\n

      Can you determine\u00a0what you would do differently if you were asked to solve equations like these?<\/p>\n

      Solve [latex]{12.5}+{ t }= {-7.5}[\/latex].<\/p>\n

      What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would solve this equation with decimals.<\/p>\n