{"id":6582,"date":"2020-10-01T16:08:13","date_gmt":"2020-10-01T16:08:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6582"},"modified":"2021-08-25T14:08:20","modified_gmt":"2021-08-25T14:08:20","slug":"1-4-solving-linear-equations-with-variables-on-both-sides-of-the-equation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/1-4-solving-linear-equations-with-variables-on-both-sides-of-the-equation\/","title":{"raw":"1.4 Solving Linear Equations with Variables on Both Sides of the Equation","rendered":"1.4 Solving Linear Equations with Variables on Both Sides of the Equation"},"content":{"raw":"
\r\n

SECTION 1.4 Learning Objectives<\/h3>\r\n1.4: Solving Linear Equations with Variables on Both Sides of the Equation<\/strong>\r\n
    \r\n \t
  • Solving linear equations with variables on both sides of the equation<\/li>\r\n \t
  • Use both the Distributive Property and combining like terms to simplify and then solve algebraic equations with variables on both sides of the equation<\/li>\r\n \t
  • Classifying solutions to linear equations<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n\r\nSome equations may have the variable on both sides of the equal sign, as in this equation: [latex]4x-6=2x+10[\/latex].\r\n

    Solving linear equations with variables on both sides of the equation<\/h2>\r\nTo 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 doesn\u2019t matter which term gets moved, [latex]4x[\/latex] or [latex]2x[\/latex], however, to avoid negative coefficients, you can move the smaller term.\r\n
    \r\n

    Example 1<\/h3>\r\nSolve: [latex]4x-6=2x+10[\/latex]\r\n\r\n[reveal-answer q=\"457216\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"457216\"]\r\n\r\nChoose the variable term to move\u2014to avoid negative terms choose [latex]2x[\/latex]\r\n

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

    Now add 6 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 2 to isolate the variable x.<\/p>\r\n

    [latex]\\begin{array}{c}\\frac{2x}{2}=\\frac{16}{2}\\\\\\\\x=8\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this video, we show an example of solving equations that have variables on both sides of the equal sign.\r\nhttps:\/\/youtu.be\/f3ujWNPL0Bw\r\n

    Use both the Distributive Property and combining like terms to simplify and then solve algebraic equations with variables on both sides of the equation<\/h2>\r\n

    Recall the Distributive Property from last section<\/h3>\r\n
    \r\n

    \u00a0The Distributive Property of Multiplication<\/h2>\r\nFor all real numbers a, b,<\/i> and c<\/i>,\u00a0[latex]a(b+c)=ab+ac[\/latex].\r\n\r\n<\/div>\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 isolate the variable<\/b>\u00a0and solve the equation.\r\n\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 (that we learned last section) twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!\r\n
    \r\n

    Example 2<\/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

    [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

    [latex]\\begin{array}{r}8t-10=-6t-3\\\\ \\underline{+6t\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+6t}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\ 14t-10=\\,\\,\\,\\,-3\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd 10 to both sides of the equation to isolate t<\/em>.\r\n

    [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 14 to completely isolate t<\/em>.\r\n

    [latex]\\begin{array}{r}14t=7\\,\\,\\,\\,\\\\\\frac{14t}{14}=\\frac{7}{14}\\end{array}[\/latex]<\/p>\r\n\r\n

    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\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\nIn the next example, we will use both the Distributive Property and combining like terms to simplify before solving the equation.\r\n
    \r\n

    ExAMPLE 3<\/h3>\r\nSolve\u00a0for [latex]x[\/latex].\r\n\r\n[latex]4\\left(x-3\\right)+2=2x-\\left(x+1\\right)[\/latex]\r\n\r\n[reveal-answer q=\"861883\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"861883\"]\r\n

    [latex]4\\left(x-3\\right)+2=2x-\\left(x+1\\right)[\/latex]<\/p>\r\nIn simplifying the right side of the equation, it may be helpful to think of [latex]-(x+1)[\/latex] as [latex]-1(x+1)[\/latex]\r\n

    [latex]4\\left(x-3\\right)+2=2x-1\\left(x+1\\right)[\/latex]<\/p>\r\nUse the distributive property to get rid of parentheses\r\n

    [latex]4x-12+2=2x-x-1[\/latex]<\/p>\r\nCombine like terms on each side of the equation\r\n

    [latex]4x-10=x-1[\/latex]<\/p>\r\nSubtract [latex]x[\/latex] from both sides of the equation\r\n

    [latex]\\begin{array}{r}4x-10=x-1\\\\ \\underline{-x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-x}\\,\\,\\,\\,\\,\\,\\,\\\\ 3x-10=\\,\\,\\,\\,-1\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd 10 to both sides of the equation to isolate the x term\r\n

    [latex]\\begin{array}{r}3x-10=-1\\\\ \\underline{+10\\,\\,\\,+10}\\\\ 3x=\\,\\,\\,9\\,\\end{array}[\/latex]<\/p>\r\nThe last step is to divide both sides by 3 to completely isolate x.\r\n

    [latex]\\begin{array}{r}3x=9\\,\\,\\,\\,\\\\\\frac{3x}{3}=\\frac{9}{3}\\end{array}[\/latex]<\/p>\r\n

    [latex]x=3[\/latex]<\/p>\r\n\r\n

    Answer<\/h4>\r\n

    [latex]x=3[\/latex]<\/p>\r\n \r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    Classifying solutions to Linear Equations<\/h2>\r\nThere are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that do not have any solutions and even some that have an infinite number of solutions. The case where an equation has no\u00a0solution is illustrated in the next example.\r\n

    Equations with No Solutions<\/h3>\r\n
    \r\n

    Example 4<\/h3>\r\nSolve for [latex]x[\/latex].\r\n\r\n[latex]12+2x\u20138=7x+5\u20135x[\/latex]\r\n\r\n[reveal-answer q=\"790409\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"790409\"]\r\n\r\nCombine like terms<\/b> on both sides of the equation.\r\n

    [latex] \\displaystyle \\begin{array}{l}12+2x-8=7x+5-5x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x+4=2x+5\\end{array}[\/latex]<\/p>\r\nIsolate the x<\/i> term by subtracting 2x<\/i> from both sides.\r\n

    [latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x+4=2x+5\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x\\,\\,\\,\\,\\,\\,\\,\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4= \\,5\\end{array}[\/latex]<\/p>\r\nThis false statement implies there are no solutions<\/strong> to this equation. Sometimes, we say the solution does not exist, or DNE for short.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice in the example above you\u00a0did not <\/i>find a value for x<\/i>. Solving for x<\/i> the way you know how, you arrive at the false statement [latex]4=5[\/latex]. Surely [latex]4[\/latex] cannot be equal to [latex]5[\/latex]!\r\n\r\nThis may make sense when you consider the second line in the solution where like terms were combined. If you multiply a number by [latex]2[\/latex] and add [latex]4[\/latex] you would never get the same answer as when you multiply that same number by [latex]2[\/latex] and add \u00a0[latex]5[\/latex]. Since there is no value of x <\/i>that will ever make this a true statement, the solution to the equation above is \u201cno solution.\u201d<\/i>\r\n\r\nBe careful that you do not confuse the solution [latex]x=0[\/latex] with \u201cno solution.\u201d The solution [latex]x=0[\/latex]\u00a0means that the value [latex]0[\/latex] satisfies the equation, so there is <\/i>a solution. \u201cNo solution\u201d means that there is no<\/strong> value, not even [latex]0[\/latex], which would satisfy the equation.\r\n\r\nAlso, be careful not to make the mistake of thinking that the equation [latex]4=5[\/latex] means that [latex]4[\/latex] and [latex]5[\/latex] are values for x<\/i> that <\/i>are solutions. If you substitute these values into the original equation, you\u2019ll see that they do not satisfy the equation. This is because there is truly no solution<\/i>\u2014there are no values for x<\/i> that will make the equation [latex]12+2x\u20138=7x+5\u20135x[\/latex]\u00a0true.\r\n

    Algebraic Equations with an Infinite Number of Solutions<\/h3>\r\nYou have seen that if an equation has no solution, you end up with a false statement instead of a value for x<\/i>. It is possible to have an equation where any value for x<\/em> will provide a solution to the equation. In the example below, notice how combining the terms [latex]5x[\/latex] and [latex]-4x[\/latex] on the left\u00a0leaves us with an equation with exactly the same terms on both sides of the equal sign.\r\n
    \r\n

    Example 5<\/h3>\r\nSolve for [latex]x[\/latex].\r\n\r\n[latex]5x+3\u20134x=3+x[\/latex]\r\n\r\n[reveal-answer q=\"773733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"773733\"]Combine like terms on both sides of the equation.\r\n

    [latex] \\displaystyle \\begin{array}{r}5x+3-4x=3+x\\\\x+3=3+x\\end{array}[\/latex]<\/p>\r\nIsolate the x<\/i> term by subtracting x<\/i> from both sides.\r\n

    [latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x+3=3+x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-x\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\,=\\,\\,3\\end{array}[\/latex]<\/p>\r\nThis true statement implies there are an infinite number of solutions to this equation, or we can also write the solution as \"All Real Numbers\"\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn that last example, we arrived at the true statement \u201c[latex]3=3[\/latex].\u201d When you end up with a true statement like this, it means that the solution to the equation is \u201call real numbers.\u201d Try substituting [latex]x=0[\/latex]\u00a0into the original equation\u2014you will get a true statement! Try [latex]x=-\\dfrac{3}{4}[\/latex] and it will also check!\r\n\r\nThis equation happens to have an infinite number of solutions. Any value for x <\/i>that you can think of will make this equation true. When you think about the context of the problem, this makes sense\u2014the equation [latex]x+3=3+x[\/latex] means \u201csome number plus [latex]3[\/latex] is equal to [latex]3[\/latex] plus that same number.\u201d We know that this is always true\u2014it is the commutative property of addition!\r\n

    \r\n

    Example 6<\/h3>\r\nSolve for [latex]x[\/latex].\r\n\r\n[latex]3\\left(2x-5\\right)=6x-15[\/latex]\r\n\r\n[reveal-answer q=\"973733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"973733\"]\r\n\r\nDistribute the [latex]3[\/latex] through the parentheses on the left-hand side.\r\n

    [latex] \\begin{array}{r}3\\left(2x-5\\right)=6x-15\\\\6x-15=6x-15\\end{array}[\/latex]<\/p>\r\nWait! This looks just like the previous example. You have the same expression on both sides of an equal sign. \u00a0No matter what number you choose for x<\/em>, you will have a true statement. We can finish the algebra:\r\n

    [latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,6x-15=6x-15\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-6x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6x\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-15\\,\\,=\\,\\,-15\\end{array}[\/latex]<\/p>\r\nThis true statement implies there are an infinite number of solutions to this equation.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more\u00a0examples of attempting to solve a linear equation with either no solution or many solutions.\r\n\r\nhttps:\/\/youtu.be\/iLkZ3o4wVxU\r\n\r\nIn the following video, we show more examples\u00a0of\u00a0solving linear equations with parentheses that have either no solution or many solutions.\r\n\r\nhttps:\/\/youtu.be\/EU_NEo1QBJ0","rendered":"

    \n

    SECTION 1.4 Learning Objectives<\/h3>\n

    1.4: Solving Linear Equations with Variables on Both Sides of the Equation<\/strong><\/p>\n

      \n
    • Solving linear equations with variables on both sides of the equation<\/li>\n
    • Use both the Distributive Property and combining like terms to simplify and then solve algebraic equations with variables on both sides of the equation<\/li>\n
    • Classifying solutions to linear equations<\/li>\n<\/ul>\n<\/div>\n

       <\/p>\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>\n

      Solving linear equations with variables on both sides of the equation<\/h2>\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 doesn\u2019t matter which term gets moved, [latex]4x[\/latex] or [latex]2x[\/latex], however, to avoid negative coefficients, you can move the smaller term.<\/p>\n

      \n

      Example 1<\/h3>\n

      Solve: [latex]4x-6=2x+10[\/latex]<\/p>\n

      Show Solution<\/span><\/p>\n
      \n

      Choose the variable term to move\u2014to avoid negative terms choose [latex]2x[\/latex]<\/p>\n

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

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

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

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

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

      In this video, we show an example of solving equations that have variables on both sides of the equal sign.
      \n