{"id":6575,"date":"2020-10-01T15:47:23","date_gmt":"2020-10-01T15:47:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6575"},"modified":"2023-01-20T03:39:19","modified_gmt":"2023-01-20T03:39:19","slug":"1-2-solving-two-step-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/1-2-solving-two-step-linear-equations\/","title":{"raw":"1.2 Solving Two-Step Linear Equations","rendered":"1.2 Solving Two-Step Linear Equations"},"content":{"raw":"
\r\n

SECTION 1.2 Learning Objective<\/h3>\r\n1.2: Solving Two-Step Linear Equations<\/strong>\r\n
    \r\n \t
  • Use properties of equality to isolate variables and solve algebraic equations<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n\r\nIn the previous section, we used the addition and multiplication properties of equality to solve algebraic equations.\u00a0 In this section, we will look at equations where both properties are needed in order to solve the equation.\r\n
    \r\n

    Addition Property of Equality<\/h3>\r\nFor all real numbers a<\/i>, b<\/i>, and c<\/i>: If [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].\r\n\r\nIf two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.\r\n\r\n<\/div>\r\n
    \r\n

    Multiplication Property of Equality<\/h3>\r\nFor all real numbers a<\/i>, b<\/i>, and c<\/i>: If a<\/i> = b<\/i>, then [latex]a\\cdot{c}=b\\cdot{c}[\/latex]\u00a0(or ab<\/i> = ac<\/i>).\r\n\r\nIf two expressions are equal to each other and you multiply both sides by the same number, the resulting expressions will also be equivalent.\r\n\r\n<\/div>\r\n

    Use properties of equality to isolate variables and solve algebraic equations<\/h2>\r\n[caption id=\"attachment_4415\" align=\"aligncenter\" width=\"300\"]\"steps Steps With an End In Sight[\/caption]\r\n\r\nThere are some equations<\/b> that you can solve in your head quickly. For example\u2014what is the value of y<\/i> in 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 6 by 2.\r\n\r\nOther equations are more complicated. Solving [latex]\\displaystyle 4\\left( \\frac{1}{3}t+\\frac{1}{2}\\right)=6[\/latex] without writing anything down is difficult! That\u2019s because this equation contains not just a variable<\/b> but also fractions and terms<\/b> inside parentheses. This is a multi-step equation<\/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.\r\n\r\nRemember 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<\/b> and the multiplication property of equality<\/b> 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\u2019ll keep both sides of the equation equal.\r\n\r\nIf the equation is in the form [latex]ax+b=c[\/latex], where x<\/i> is the variable, you can solve the equation as before. First \u201cundo\u201d the addition and subtraction, and then \u201cundo\u201d the multiplication and division.\r\n
    \r\n

    Example 1<\/h3>\r\nSolve [latex]3y+2=11[\/latex].\r\n\r\n[reveal-answer q=\"843520\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"843520\"]\r\n\r\nSubtract 2 from both sides of the equation to get the term with the variable by itself.\r\n

    [latex] \\displaystyle \\begin{array}{r}3y+2\\,\\,\\,=\\,\\,11\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,-2\\,\\,\\,\\,\\,\\,\\,\\,-2}\\\\3y\\,\\,\\,\\,=\\,\\,\\,\\,\\,9\\end{array}[\/latex]<\/p>\r\nDivide both sides of the equation by 3 to get a coefficient of 1 for the variable.\r\n

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

    Answer<\/h4>\r\n[latex]y=3[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn this next video you will see a few more examples of solving two-step linear equations.\r\n\r\nhttps:\/\/youtu.be\/9ITsXICV2u0\r\n\r\n ","rendered":"
    \n

    SECTION 1.2 Learning Objective<\/h3>\n

    1.2: Solving Two-Step Linear Equations<\/strong><\/p>\n

      \n
    • Use properties of equality to isolate variables and solve algebraic equations<\/li>\n<\/ul>\n<\/div>\n

       <\/p>\n

      In the previous section, we used the addition and multiplication properties of equality to solve algebraic equations.\u00a0 In this section, we will look at equations where both properties are needed in order to solve the equation.<\/p>\n

      \n

      Addition Property of Equality<\/h3>\n

      For all real numbers a<\/i>, b<\/i>, and c<\/i>: If [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex].<\/p>\n

      If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.<\/p>\n<\/div>\n

      \n

      Multiplication Property of Equality<\/h3>\n

      For all real numbers a<\/i>, b<\/i>, and c<\/i>: If a<\/i> = b<\/i>, then [latex]a\\cdot{c}=b\\cdot{c}[\/latex]\u00a0(or ab<\/i> = ac<\/i>).<\/p>\n

      If two expressions are equal to each other and you multiply both sides by the same number, the resulting expressions will also be equivalent.<\/p>\n<\/div>\n

      Use properties of equality to isolate variables and solve algebraic equations<\/h2>\n
      \"steps<\/p>\n

      Steps With an End In Sight<\/p>\n<\/div>\n

      There are some equations<\/b> that you can solve in your head quickly. For example\u2014what is the value of y<\/i> in 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 6 by 2.<\/p>\n

      Other equations are more complicated. Solving [latex]\\displaystyle 4\\left( \\frac{1}{3}t+\\frac{1}{2}\\right)=6[\/latex] without writing anything down is difficult! That\u2019s because this equation contains not just a variable<\/b> but also fractions and terms<\/b> inside parentheses. This is a multi-step equation<\/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<\/b> and the multiplication property of equality<\/b> 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\u2019ll keep both sides of the equation equal.<\/p>\n

      If the equation is in the form [latex]ax+b=c[\/latex], where x<\/i> is the variable, you can solve the equation as before. First \u201cundo\u201d the addition and subtraction, and then \u201cundo\u201d the multiplication and division.<\/p>\n

      \n

      Example 1<\/h3>\n

      Solve [latex]3y+2=11[\/latex].<\/p>\n

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

      Subtract 2 from both sides of the equation to get the term with the variable by itself.<\/p>\n

      [latex]\\displaystyle \\begin{array}{r}3y+2\\,\\,\\,=\\,\\,11\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,-2\\,\\,\\,\\,\\,\\,\\,\\,-2}\\\\3y\\,\\,\\,\\,=\\,\\,\\,\\,\\,9\\end{array}[\/latex]<\/p>\n

      Divide both sides of the equation by 3 to get a coefficient of 1 for the variable.<\/p>\n

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

      Answer<\/h4>\n

      [latex]y=3[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n

      In this next video you will see a few more examples of solving two-step linear equations.<\/p>\n