{"id":1269,"date":"2023-04-04T19:19:03","date_gmt":"2023-04-04T19:19:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=1269"},"modified":"2025-08-24T04:12:44","modified_gmt":"2025-08-24T04:12:44","slug":"algebraic-equations-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/algebraic-equations-learn-it-3\/","title":{"raw":"Algebraic Equations: Learn It 3","rendered":"Algebraic Equations: Learn It 3"},"content":{"raw":"

Multi-Step Equations<\/h2>\r\n

An equation will always contain an equal sign with an expression on each side. Think of an equal sign as meaning \"the same as.\" Some examples of equations are [latex]y = mx +b[\/latex], [latex]\\Large\\frac{3}{4}\\normalsize r = v^{3} - r[\/latex], and [latex]2(6-d) + f(3 +k) =\\Large\\frac{1}{4}\\normalsize d[\/latex].<\/p>\r\n

The following figure shows how coefficients, variables, terms, and expressions all come together to make equations. In the equation [latex]2x-3^2=10x[\/latex], the variable is [latex]x[\/latex], a coefficient is [latex]10[\/latex], a term is [latex]10x[\/latex], an expression is [latex]2x-3^2[\/latex].<\/p>\r\n

\r\n[caption id=\"attachment_4693\" align=\"aligncenter\" width=\"424\"]\"Equation Figure 1. This is an equation with variables, an expression, coefficients, and terms[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

There are some equations that you can solve in your head quickly. For example, what is the value of [latex]y[\/latex]<\/i>\u00a0in the equation [latex]2y=6[\/latex]? Chances are you didn\u2019t need to get out a pencil and paper to calculate that [latex]y=3[\/latex]. You only needed to do one thing to get the answer: divide [latex]6[\/latex] by [latex]2[\/latex].<\/p>\r\n

Other equations are more complicated. Solving [latex]\\displaystyle 4\\left(\\frac{1}{3}\\normalsize t+\\frac{1}{2}\\normalsize\\right)=6[\/latex] without writing anything down is difficult! That is because this equation contains not just a variable but also fractions and terms inside parentheses. This is a multi-step equation,\u00a0<\/b>one that takes several steps to solve. Although multi-step equations take more time and more operations, they can still be simplified and solved by applying basic algebraic rules.<\/p>\r\n

Remember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The [pb_glossary id=\"13469\"]addition property of equality[\/pb_glossary] and the [pb_glossary id=\"13470\"]multiplication property of equality[\/pb_glossary] explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you will keep both sides of the equation equal.<\/p>\r\n

\r\n

How To: Solving Multi-Step Equations<\/strong><\/p>\r\n

    \r\n\t
  1. (Optional) Multiply to clear any fractions or decimals.<\/li>\r\n\t
  2. Simplify each side by clearing parentheses and combining like terms.<\/li>\r\n\t
  3. Add or subtract to isolate the variable term\u2014you may have to move a term with the variable.<\/li>\r\n\t
  4. Multiply or divide to isolate the variable.<\/li>\r\n\t
  5. Check the solution.<\/li>\r\n<\/ol>\r\n<\/section>\r\n
    Solve the following for [latex]x[\/latex]:
    [latex]3x+5x+4-x+7=88[\/latex]<\/center>[reveal-answer q=\"455516\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"455516\"]There are three like terms involving a variable: [latex]3x[\/latex], [latex]5x[\/latex], and [latex]\u2013x[\/latex]. Combine these like terms. [latex]4[\/latex] and [latex]7[\/latex] are also like terms and can be added.\r\n\r\n

    [latex]\\begin{array}{r}\\,\\,3x+5x+4-x+7=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x+11=\\,\\,\\,88\\end{array}[\/latex]<\/p>\r\n

    The equation is now in the form [latex]ax+b=c[\/latex], so we can solve as before.<\/p>\r\n

    Subtract [latex]11[\/latex] from both sides.<\/p>\r\n

    [latex]\\begin{array}{r}7x+11\\,\\,\\,=\\,\\,\\,88\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-11\\,\\,\\,\\,\\,\\,\\,-11}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7x\\,\\,\\,=\\,\\,\\,77\\end{array}[\/latex]<\/p>\r\n

    Divide both sides by [latex]7[\/latex].<\/p>\r\n

    [latex]\\begin{array}{r}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{7x}\\,\\,\\,=\\,\\,\\,\\underline{77}\\\\7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,=\\,\\,\\,11\\end{array}[\/latex]<\/p>\r\n

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

    Some equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[\/latex].<\/p>\r\n

    To solve this equation, we need to \u201cmove\u201d one of the variable terms. This can make it difficult to decide which side to work with. It does not matter which term gets moved, [latex]4x[\/latex] or [latex]2x[\/latex]; however, to avoid negative coefficients, you can move the smaller term.<\/p>\r\n

    Solve the following for [latex]x[\/latex]:
    [latex]4x-6=2x+10[\/latex]<\/center>[reveal-answer q=\"457216\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"457216\"]Choose the variable term to move\u2014to avoid negative terms choose [latex]2x[\/latex].\r\n\r\n

    [latex]\\begin{array}{r}4x-6=2x+10\\,\\\\\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\2x-6=10\\end{array}[\/latex]<\/p>\r\n

    <\/center>\r\n

    Now add [latex]6[\/latex] to both sides to isolate the term with the variable.<\/p>\r\n

    [latex]\\begin{array}{r}2x-6=10\\\\\\underline{\\,\\,\\,\\,+6\\,\\,\\,+6}\\\\2x=16\\end{array}[\/latex]<\/p>\r\n

    Now divide each side by [latex]2[\/latex] to isolate the variable [latex]x[\/latex].<\/p>\r\n

    [latex]\\begin{array}{c}\\Large\\frac{2x}{2}\\normalsize=\\Large\\frac{16}{2}\\\\\\\\\\normalsize{x=8}\\end{array}[\/latex]<\/p>\r\n

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

    Solving Multi-Step Equations With Absolute Value<\/h3>\r\n

    We can apply the same techniques we used for solving a one-step equation that contain an absolute value to an equation that will take more than one step to solve.<\/p>\r\n

    Let's recall the steps to solving one-step equation that contains an absolute value.\r\n\r\n
      \r\n\t
    1. Isolate the absolute value: Move all terms except the absolute value to one side of the equation. Then, isolate the absolute value on one side of the equation by adding or subtracting the same value from both sides.<\/li>\r\n\t
    2. Remove the absolute value: Once the absolute value is isolated, we can remove it by splitting the equation into two cases: one where the expression inside the absolute value is positive and one where it is negative. Then, solve each equation separately.<\/li>\r\n\t
    3. Check the solution: After obtaining the solutions, we should check them by plugging them back into the original equation to make sure they are valid. If the solution satisfies the original equation, then it is the correct solution.<\/li>\r\n<\/ol>\r\n<\/section>\r\n

      Let's practice a simple algebraic equation that contains an absolute value.<\/p>\r\n

      Solve the following for [latex]p[\/latex]:
      [latex]\\left|2p\u20134\\right|=26[\/latex]<\/center>[reveal-answer q=\"371950\"]Show Solution[\/reveal-answer] [hidden-answer a=\"371950\"]Write the two equations that will give an absolute value of [latex]26[\/latex].\r\n\r\n

      [latex] \\displaystyle 2p-4=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,2p-4=\\,-26[\/latex]<\/p>\r\n

      Solve each equation for [latex]p[\/latex] by isolating the variable.<\/i><\/p>\r\n

      [latex] \\displaystyle \\begin{array}{r}2p-4=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2p-4=\\,-26\\\\\\underline{\\,\\,\\,\\,\\,\\,+4\\,\\,\\,\\,+4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,+4\\,\\,\\,\\,\\,\\,\\,+4}\\\\\\underline{2p}\\,\\,\\,\\,\\,\\,=\\underline{30}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{2p}\\,\\,\\,\\,\\,=\\,\\underline{-22}\\\\2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,p=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,p=\\,-11\\end{array}[\/latex]<\/p>\r\n

      Check the solutions in the original equation.<\/p>\r\n

      [latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,\\left| 2p-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 2p-4 \\right|=26\\\\\\left| 2(15)-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\left| 2(-11)-4 \\right|=26\\\\\\,\\,\\,\\,\\,\\left| 30-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -22-4 \\right|=26\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 26 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -26 \\right|=26\\end{array}[\/latex]<\/p>\r\n

      Both solutions check!<\/p>\r\n

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

      Now let us look at an example where you need to do an algebraic step or two before you can write your two equations. The goal here is to get the absolute value on one side of the equation by itself. Then we can proceed as we did in the previous example.<\/p>\r\n

      Solve the following for [latex]w[\/latex]:
      [latex]3\\left|4w\u20131\\right|\u20135=10[\/latex]<\/center>[reveal-answer q=\"303228\"]Show Solution[\/reveal-answer] [hidden-answer a=\"303228\"]Isolate the term with the absolute value by adding [latex]5[\/latex] to both sides.\r\n\r\n

      [latex]\\begin{array}{r}3\\left|4w-1\\right|-5=10\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+5\\,\\,\\,+5}\\\\ 3\\left|4w-1\\right|=15\\end{array}[\/latex]<\/p>\r\n

      Divide both sides by [latex]3[\/latex]. Now the absolute value is isolated.<\/p>\r\n

      [latex]\\begin{array}{r} \\underline{3\\left|4w-1\\right|}=\\underline{15}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\,\\\\\\left|4w-1\\right|=\\,\\,5\\end{array}[\/latex]<\/p>\r\n

      Write the two equations that will give an absolute value of [latex]5[\/latex] and solve them.<\/p>\r\n

      [latex] \\displaystyle \\begin{array}{r}4w-1=5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4w-1=-5\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,+1\\,\\,+1}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,+1\\,\\,\\,\\,\\,+1}\\\\\\,\\,\\,\\,\\,\\underline{4w}=\\underline{6}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{4w}\\,\\,\\,\\,\\,\\,\\,=\\underline{-4}\\\\4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,w=\\Large\\frac{3}{2}\\normalsize \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,w=-1\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,w=\\Large\\frac{3}{2}\\normalsize \\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n

      Check the solutions in the original equation.<\/p>\r\n

      [latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\\\\\\\3\\left| 4\\left(\\Large\\frac{3}{2}\\normalsize\\right)-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\\\\\\\\\,\\,\\,\\,\\,\\,3\\left|\\Large\\frac{12}{2}\\normalsize -1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,3\\left| 4(-1)-1\\, \\right|-5=10\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 6-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| -4-1\\, \\right|-5=10\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left(5\\right)-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| -5 \\right|-5=10\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15-5=10\\\\10=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,10=10\\end{array}[\/latex]<\/p>\r\n

      Both solutions check!<\/p>\r\n

      [\/hidden-answer]<\/p>\r\n<\/section>","rendered":"

      Multi-Step Equations<\/h2>\n

      An equation will always contain an equal sign with an expression on each side. Think of an equal sign as meaning “the same as.” Some examples of equations are [latex]y = mx +b[\/latex], [latex]\\Large\\frac{3}{4}\\normalsize r = v^{3} - r[\/latex], and [latex]2(6-d) + f(3 +k) =\\Large\\frac{1}{4}\\normalsize d[\/latex].<\/p>\n

      The following figure shows how coefficients, variables, terms, and expressions all come together to make equations. In the equation [latex]2x-3^2=10x[\/latex], the variable is [latex]x[\/latex], a coefficient is [latex]10[\/latex], a term is [latex]10x[\/latex], an expression is [latex]2x-3^2[\/latex].<\/p>\n

      \n
      \"Equation
      Figure 1. This is an equation with variables, an expression, coefficients, and terms<\/figcaption><\/figure>\n<\/div>\n

       <\/p>\n

      There are some equations that you can solve in your head quickly. For example, what is the value of [latex]y[\/latex]<\/i>\u00a0in the equation [latex]2y=6[\/latex]? Chances are you didn\u2019t need to get out a pencil and paper to calculate that [latex]y=3[\/latex]. You only needed to do one thing to get the answer: divide [latex]6[\/latex] by [latex]2[\/latex].<\/p>\n

      Other equations are more complicated. Solving [latex]\\displaystyle 4\\left(\\frac{1}{3}\\normalsize t+\\frac{1}{2}\\normalsize\\right)=6[\/latex] without writing anything down is difficult! That is because this equation contains not just a variable but also fractions and terms inside parentheses. This is a multi-step equation,\u00a0<\/b>one that takes several steps to solve. Although multi-step equations take more time and more operations, they can still be simplified and solved by applying basic algebraic rules.<\/p>\n

      Remember that you can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The addition property of equality<\/a> and the multiplication property of equality<\/a> explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you will keep both sides of the equation equal.<\/p>\n

      \n

      How To: Solving Multi-Step Equations<\/strong><\/p>\n

        \n
      1. (Optional) Multiply to clear any fractions or decimals.<\/li>\n
      2. Simplify each side by clearing parentheses and combining like terms.<\/li>\n
      3. Add or subtract to isolate the variable term\u2014you may have to move a term with the variable.<\/li>\n
      4. Multiply or divide to isolate the variable.<\/li>\n
      5. Check the solution.<\/li>\n<\/ol>\n<\/section>\n
        Solve the following for [latex]x[\/latex]:<\/p>\n
        [latex]3x+5x+4-x+7=88[\/latex]<\/div>\n