{"id":6580,"date":"2020-10-01T16:03:52","date_gmt":"2020-10-01T16:03:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6580"},"modified":"2023-03-10T00:47:56","modified_gmt":"2023-03-10T00:47:56","slug":"1-3-solving-linear-equations-using-the-distributive-property","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/1-3-solving-linear-equations-using-the-distributive-property\/","title":{"raw":"1.3 Simplifying and Solving Linear Equations","rendered":"1.3 Simplifying and Solving Linear Equations"},"content":{"raw":"
\r\n

SECTION 1.3 Learning Objectives<\/h3>\r\n1.3: Simplifying and Solving Linear Equations<\/strong>\r\n
    \r\n \t
  • Apply the Distributive Property to simplify and then solve algebraic equations<\/li>\r\n \t
  • Combine like terms to simplify and then solve algebraic equations<\/li>\r\n \t
  • Use both the Distributive Property and combining like terms to simplify and then solve algebraic equations<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n

    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, if not impossible. To get rid of these unwanted parentheses we have the distributive property. Using this property we multiply the number in front of the parentheses by each term inside of the parentheses.\r\n
    \r\n

    The Distributive Property of Multiplication<\/h3>\r\nFor all real numbers a, b,<\/i> and c<\/i>,\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 isolate the variable<\/b>\u00a0and solve the equation.\r\n\r\n<\/div>\r\nThe video below will review some examples of this distributive property in action.\r\n\r\nhttps:\/\/youtu.be\/LdhKEtTrJ60\r\n

    Apply the Distributive Property to simplify and then solve algebraic equations<\/h2>\r\n
    \r\n

    Example 1<\/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

    [latex]\\begin{array}{r}4\\left(2a+3\\right)=28\\\\ 8a+12=28\\end{array}[\/latex]<\/p>\r\nSubtract 12\u00a0from both sides to isolate\u00a0the variable term.\r\n

    [latex]\\begin{array}{r}8a+12\\,\\,\\,=\\,\\,\\,28\\\\ \\underline{-12\\,\\,\\,\\,\\,\\,-12}\\\\ 8a\\,\\,\\,=\\,\\,\\,16\\end{array}[\/latex]<\/p>\r\nDivide both terms by 8 to get a coefficient of 1.\r\n

    [latex]\\begin{array}{r}\\underline{8a}=\\underline{16}\\\\8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\\\a\\,=\\,\\,2\\end{array}[\/latex]<\/p>\r\n\r\n

    Answer<\/h4>\r\n[latex]a=2[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n

    Combine like terms to simplify and then solve algebraic equations<\/h2>\r\nSometimes we need to combine like terms on one or both sides of the equations to simplify before solving the equation.\u00a0 Like terms are terms with the same variable part.\u00a0 Constant terms are also considered like terms.\r\n
    \r\n

    Example 2<\/h3>\r\nSolve: [latex]7x-9-2x=-5+11[\/latex]\r\n\r\n[reveal-answer q=\"195358\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"195358\"]\r\n

    [latex]7x-9-2x=-5+11[\/latex]<\/p>\r\nCombine the [latex]x[\/latex] terms on the left and the constant terms on the right.\r\n

    [latex]5x-9=6[\/latex]<\/p>\r\nAdd [latex]9[\/latex] to both sides of the equation.\r\n

    [latex]\\begin{array}{r}5x-9\\,\\,\\,=\\,\\,\\,6\\\\ \\underline{+9\\,\\,\\,\\,\\,\\,+9}\\\\ 5x\\,\\,\\,=\\,\\,\\,15\\end{array}[\/latex]<\/p>\r\nDivide both sides of the equation with 5\r\n

    [latex]\\begin{array}{r}\\underline{5x}=\\underline{15}\\\\5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,5\\\\x\\,=\\,\\,3\\end{array}[\/latex]<\/p>\r\nAnswer: [latex]x=3[\/latex]\r\n\r\n \r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe video below will show some more examples of combining like terms to solve an equation\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=IEcAenPKFq0&feature=youtu.be[\/embed]\r\n

    Use both the Distributive Property and combining like terms to simplify and then solve algebraic equations<\/h2>\r\nIn the next example, we will use both the Distributive Property and combining like terms to simply before solving the equation.\r\n
    \r\n

    Example 3<\/h3>\r\nSolve: [latex]5-14=3-2(x+1)+7x[\/latex]\r\n\r\n[reveal-answer q=\"588062\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"588062\"]\r\n

    [latex]5-14=3-2(x+1)+7x[\/latex]<\/p>\r\nApply the distributive property to expand [latex]-2\\left(x+1\\right)[\/latex] to [latex]-2x-2[\/latex]\r\n

    [latex]\\begin{array}{r}5-14=3-2x-2+7x\\end{array}[\/latex]<\/p>\r\nCombine like terms on each side of the equation\r\n

    [latex]-9=5x+1[\/latex]<\/p>\r\nSubtract [latex]1[\/latex] from both sides of the equation\r\n

    [latex]\\begin{array}{r}-9\\,\\,\\,=\\,\\,\\,5x+1\\\\ \\underline{-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-1}\\\\ -10\\,\\,\\,=\\,\\,\\,5x\\end{array}[\/latex]<\/p>\r\nDivide both sides of the equation by [latex]5[\/latex]\r\n

    [latex]\\begin{array}{r}\\underline{-10}=\\underline{5x}\\\\5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,5\\\\-2\\,=\\,\\,x\\end{array}[\/latex]<\/p>\r\nAnswer: [latex]x=-2[\/latex]\r\n\r\n \r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe video below will show some more examples of this type:\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=e-xi6fbPp-8&feature=youtu.be[\/embed]","rendered":"

    \n

    SECTION 1.3 Learning Objectives<\/h3>\n

    1.3: Simplifying and Solving Linear Equations<\/strong><\/p>\n

      \n
    • Apply the Distributive Property to simplify and then solve algebraic equations<\/li>\n
    • Combine like terms to simplify and then solve algebraic equations<\/li>\n
    • Use both the Distributive Property and combining like terms to simplify and then solve algebraic equations<\/li>\n<\/ul>\n<\/div>\n

       <\/p>\n

      The Distributive Property<\/h2>\n

      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

      Parentheses can\u00a0make solving a problem difficult, if not impossible. To get rid of these unwanted parentheses we have the distributive property. Using this property we multiply the number in front of the parentheses by each term inside of the parentheses.<\/p>\n

      \n

      The Distributive Property of Multiplication<\/h3>\n

      For all real numbers a, b,<\/i> and c<\/i>,\u00a0[latex]a(b+c)=ab+ac[\/latex].<\/p>\n

      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 isolate the variable<\/b>\u00a0and solve the equation.<\/p>\n<\/div>\n

      The video below will review some examples of this distributive property in action.<\/p>\n