{"id":6564,"date":"2020-10-01T14:51:31","date_gmt":"2020-10-01T14:51:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6564"},"modified":"2026-01-12T21:18:31","modified_gmt":"2026-01-12T21:18:31","slug":"1-1-solving-one-step-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/1-1-solving-one-step-linear-equations\/","title":{"raw":"1.1 Solving One-Step Linear Equations","rendered":"1.1 Solving One-Step Linear Equations"},"content":{"raw":"
\r\n

SECTION 1.1 Learning Objectives<\/h3>\r\n1.1: Solving One-Step Linear Equations<\/strong>\r\n
    \r\n \t
  • Solve algebraic equations using the Addition Property of Equality<\/li>\r\n \t
  • Solve algebraic equations using the Multiplication Property of Equality<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n\r\nFirst, let's define some important terminology:<\/span>\r\n
      \r\n \t
    • variables:\u00a0<\/strong> variables are symbols that stand for an unknown quantity, they are often represented with letters, like <\/span>x<\/i>, <\/span>y<\/i>, or <\/span>z<\/i>.<\/span><\/li>\r\n \t
    • coefficient:\u00a0<\/strong>Sometimes a variable is multiplied by a number. This number is called the coefficient of the variable. For example, the coefficient of 3x <\/i>is 3.<\/li>\r\n \t
    • term:\u00a0<\/strong>a single number, or variables and numbers connected by multiplication. -4, 6x and [latex]x^2[\/latex] are all terms<\/li>\r\n \t
    • expression: <\/strong>groups of terms connected by addition and subtraction.\u00a0 [latex]2x^2-5[\/latex] is an expression<\/li>\r\n \t
    • equation: <\/strong>\u00a0an equation is a mathematical statement that two expressions are equal. An equation will always contain an equal sign with an expression on each side.\u00a0Think of an equal sign as meaning \"the same as.\" Some examples of equations are\u00a0[latex]y = mx +b[\/latex], \u00a0[latex]\\frac{3}{4}r = v^{3} - r[\/latex], and \u00a0[latex]2(6-d) + f(3 +k) = \\frac{1}{4}d[\/latex]<\/li>\r\n<\/ul>\r\nThe 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].\r\n\r\n[caption id=\"attachment_4693\" align=\"aligncenter\" width=\"424\"]\"Equation: Equation made of coefficients, variables, terms and expressions.[\/caption]\r\n

      <\/h3>\r\n

      Solve algebraic equations using the Addition Property of Equality<\/span><\/h3>\r\nAn important property of equations is one that states that you can add the same quantity to both sides of an equation and still maintain an equivalent equation. Sometimes people refer to this as keeping the equation \u201cbalanced.\u201d If you think of an equation as being like a balance scale, the quantities on each side of the equation are equal, or balanced.\r\n\r\nLet\u2019s look at a simple numeric equation, [latex]3+7=10[\/latex], to explore the idea of an equation as being balanced.\r\n\r\n\"A\r\n\r\nThe expressions on each side of the equal sign are equal, so you can add the same value to each side and maintain the equality. Let\u2019s see what happens when 5 is added to each side.\r\n

      [latex]3+7+5=10+5[\/latex]<\/p>\r\nSince each expression is equal to 15, you can see that adding 5 to each side of the original equation resulted in a true equation. The equation is still \u201cbalanced.\u201d\r\n\r\nOn the other hand, let\u2019s look at what would happen if you added 5 to only one side of the equation.\r\n

      [latex]\\begin{array}{r}3+7=10\\\\3+7+5=10\\\\15\\neq 10\\end{array}[\/latex]<\/p>\r\nAdding 5 to only one side of the equation resulted in an equation that is false. The equation is no longer \u201cbalanced,\u201d and it is no longer a true 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

      <\/h3>\r\nWhen you solve an equation, you find the value of the variable that makes the equation true. In order to solve the equation, you isolate the variable<\/b>. Isolating the variable means rewriting an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.\r\n\r\nWhen the equation involves addition or subtraction, use the inverse operation to \u201cundo\u201d the operation in order to isolate the variable. For addition and subtraction, your goal is to change any value being added or subtracted to 0, the additive identity.\r\n\r\nIn the following simulation, you can adjust the quantity being added or subtracted to each side of an equation to see how important it is to perform the same operation on both sides of an equation when you are solving.\r\n
      \r\n

      Example 1<\/h3>\r\nSolve [latex]x-6=8[\/latex].\r\n\r\n[reveal-answer q=\"577240\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"577240\"]\r\n\r\nThis equation means that if you begin with some unknown number, x<\/i>, and subtract 6, you will end up with 8. You are trying to figure out the value of the variable x.<\/i>\r\n\r\nUsing the Addition Property of Equality, add 6 to both sides of the equation to isolate the variable. You choose to add 6 because 6 is being subtracted from the variable.\r\n

      [latex] \\displaystyle \\begin{array}{r}x-6\\,\\,\\,=\\,\\,\\,\\,8\\\\\\,\\,\\,\\,\\,\\,\\,\\underline{+\\,6\\,\\,\\,\\,\\,\\,\\,\\,+6}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,=\\, 14\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n[latex]x=14[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n
      \r\n\r\nSolve [latex]x+5=27[\/latex].\r\n\r\n[reveal-answer q=\"579240\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"579240\"]\r\n\r\nThis equation means that if you begin with some unknown number, x<\/i>, and add 5, you will end up with 27. You are trying to figure out the value of the variable x.<\/i>\r\n\r\nUsing the Addition Property of Equality, subtract 5 from both sides of the equation to isolate the variable. You choose to subtract 5, as 5 is being added from the variable.\r\n

      [latex] \\displaystyle \\begin{array}{r}x+5\\,\\,=\\,\\,27\\\\\\,\\,\\,\\,\\,\\,\\,\\underline{-5\\,\\,\\,\\,\\,\\,\\,\\,-5}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,=\\, 22\\end{array}[\/latex]<\/p>\r\n\r\n

      Answer<\/h4>\r\n[latex]x=22[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video two examples of using the addition property of equality are shown.\r\nhttps:\/\/youtu.be\/VsWrFKFerSY\r\nSince subtraction can be written as addition (adding the opposite), the addition property of equality<\/b> can be used for subtraction as well. So just as you can add the same value to each side of an equation without changing the meaning of the equation, you can subtract the same value from each side of an equation.\r\n
      \r\n

      Example 2<\/h3>\r\nSolve [latex]x+10=-65[\/latex]. Check your solution.\r\n\r\n[reveal-answer q=\"684455\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"684455\"]\r\n\r\nSolve:\r\n

      [latex]x+10=-65[\/latex]<\/p>\r\nSince 10 is being added to the variable, subtract 10 from both sides. Note that subtracting 10 is the same as adding [latex]\u201310[\/latex].\r\n

      [latex] \\displaystyle \\begin{array}{r}x+10\\,\\,=\\,\\,\\,\\,-65\\\\\\,\\,\\,\\,\\,\\underline{-10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-10}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,=\\,\\,\\,-75\\end{array}[\/latex]<\/p>\r\nTo check, substitute the solution, [latex]\u201375[\/latex] for x <\/i>in the original equation, then simplify.\r\n

      [latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,x+10\\,\\,\\,=-65\\\\-75+\\,10\\,\\,\\,=-65\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-65\\,\\,\\,=-65\\end{array}[\/latex]<\/p>\r\nThis equation is true, so the solution is correct.\r\n

      Answer<\/h4>\r\n[latex]x=\u201375[\/latex] is the solution to the equation [latex]x+10=\u201365[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n
      \r\n\r\nSolve [latex]x-4=-32[\/latex]. Check your solution.\r\n\r\n[reveal-answer q=\"624455\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624455\"]\r\n\r\nSolve:\r\n

      [latex]x-4=-32[\/latex]<\/p>\r\nSince 4 is being subtracted from\u00a0the variable, add 4 to\u00a0both sides.\r\n

      [latex] \\displaystyle \\begin{array}{r}x-4\\,\\,=\\,\\,\\,\\,-32\\\\\\,\\,\\,\\,\\,\\underline{+4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+4}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,=\\,\\,\\,-28\\end{array}[\/latex]<\/p>\r\nCheck:\r\n\r\nTo check, substitute the solution, [latex]\u201328[\/latex] for x <\/i>in the original equation, then simplify.\r\n

      [latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,x-4\\,\\,\\,=-32\\\\-28-\\,4\\,\\,\\,=-32\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-32\\,\\,\\,=-32\\end{array}[\/latex]<\/p>\r\nThis equation is true, so the solution is correct.\r\n

      Answer<\/h4>\r\n[latex]x=\u201328[\/latex] is the solution to the equation [latex]x-4=\u201332[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIt is always a good idea to check your answer whether you are\u00a0requested to or not.\r\n\r\nThe following video presents two examples of using the addition property of equality when there are negative integers in the equation.\r\n\r\nhttps:\/\/youtu.be\/D3T8eCT5U_w\r\n

      Solve algebraic equations using the Multiplication Property of Equality<\/h2>\r\nJust as you can add or subtract the same exact quantity on both sides of an equation, you can also multiply both sides of an equation by the same quantity to write an equivalent equation. Let\u2019s look at a numeric equation, [latex]5\\cdot3=15[\/latex], to start. If you multiply both sides of this equation by 2, you will still have a true equation.\r\n\r\n[latex]\\begin{array}{r}5\\cdot 3=15\\,\\,\\,\\,\\,\\,\\, \\\\ 5\\cdot3\\cdot2=15\\cdot2 \\\\ 30=30\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]\r\n\r\nThis characteristic of equations is generalized in the M<\/strong>ultiplication Property of Equality<\/b>.\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\nWhen the equation involves multiplication or division, you can \u201cundo\u201d these operations by using the inverse operation to isolate the variable. When the operation is multiplication or division, your goal is to change the coefficient to 1, the multiplicative identity.\r\n
      \r\n

      Example 3<\/h3>\r\nSolve [latex]3x=24[\/latex]. When you are done, check your solution.\r\n\r\n[reveal-answer q=\"42404\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"42404\"]Divide both sides of the equation by 3 to isolate the variable (have a coefficient of 1).\u00a0Dividing by 3 is the same as having multiplied by [latex] \\frac{1}{3}[\/latex].\r\n

      [latex]\\begin{array}{r}\\underline{3x}=\\underline{24}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\\\x=8\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nCheck by substituting your solution, 8, for the variable in the original equation.\r\n

      [latex]\\begin{array}{r}3x=24 \\\\ 3\\cdot8=24 \\\\ 24=24\\end{array}[\/latex]<\/p>\r\nThe solution is correct!\r\n

      Answer<\/h4>\r\n[latex]x=8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIf the equation involves division, you can multiply the coefficient by the multiplicative inverse (reciprocal) in order to change the coefficient to 1. See the next problem to see an example of this.\r\n
      \r\n

      Example 4<\/h3>\r\nSolve [latex] \\frac{x}{2}={ 8}[\/latex] for x.\r\n[reveal-answer q=\"128018\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"128018\"]The difference between this and the previous example is that the x is being divided by a number, instead of multiplied. In the last example we divided by 3 to isolate the x<\/em>. In this problem we will multiply both sides by 2, which will change the coefficient on the x to a 1.\r\n

      [latex] \\frac{x}{2}={ 8}[\/latex]<\/p>\r\nMultiply both sides by 2:\r\n

      [latex]\\left(2\\right)\\frac{x}{2}=\\left(2\\right){ 8}[\/latex]<\/p>\r\n

      [latex]\\frac{2x}{2} = 16[\/latex]<\/p>\r\n

      [latex]1x=16[\/latex]<\/p>\r\n

      [latex]x=16[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"

      \n

      SECTION 1.1 Learning Objectives<\/h3>\n

      1.1: Solving One-Step Linear Equations<\/strong><\/p>\n

        \n
      • Solve algebraic equations using the Addition Property of Equality<\/li>\n
      • Solve algebraic equations using the Multiplication Property of Equality<\/li>\n<\/ul>\n<\/div>\n

         <\/p>\n

        First, let’s define some important terminology:<\/span><\/p>\n

          \n
        • variables:\u00a0<\/strong> variables are symbols that stand for an unknown quantity, they are often represented with letters, like <\/span>x<\/i>, <\/span>y<\/i>, or <\/span>z<\/i>.<\/span><\/li>\n
        • coefficient:\u00a0<\/strong>Sometimes a variable is multiplied by a number. This number is called the coefficient of the variable. For example, the coefficient of 3x <\/i>is 3.<\/li>\n
        • term:\u00a0<\/strong>a single number, or variables and numbers connected by multiplication. -4, 6x and [latex]x^2[\/latex] are all terms<\/li>\n
        • expression: <\/strong>groups of terms connected by addition and subtraction.\u00a0 [latex]2x^2-5[\/latex] is an expression<\/li>\n
        • equation: <\/strong>\u00a0an equation is a mathematical statement that two expressions are equal. An equation will always contain an equal sign with an expression on each side.\u00a0Think of an equal sign as meaning “the same as.” Some examples of equations are\u00a0[latex]y = mx +b[\/latex], \u00a0[latex]\\frac{3}{4}r = v^{3} - r[\/latex], and \u00a0[latex]2(6-d) + f(3 +k) = \\frac{1}{4}d[\/latex]<\/li>\n<\/ul>\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

          \"Equation:<\/p>\n

          Equation made of coefficients, variables, terms and expressions.<\/p>\n<\/div>\n

          <\/h3>\n

          Solve algebraic equations using the Addition Property of Equality<\/span><\/h3>\n

          An important property of equations is one that states that you can add the same quantity to both sides of an equation and still maintain an equivalent equation. Sometimes people refer to this as keeping the equation \u201cbalanced.\u201d If you think of an equation as being like a balance scale, the quantities on each side of the equation are equal, or balanced.<\/p>\n

          Let\u2019s look at a simple numeric equation, [latex]3+7=10[\/latex], to explore the idea of an equation as being balanced.<\/p>\n

          \"A<\/p>\n

          The expressions on each side of the equal sign are equal, so you can add the same value to each side and maintain the equality. Let\u2019s see what happens when 5 is added to each side.<\/p>\n

          [latex]3+7+5=10+5[\/latex]<\/p>\n

          Since each expression is equal to 15, you can see that adding 5 to each side of the original equation resulted in a true equation. The equation is still \u201cbalanced.\u201d<\/p>\n

          On the other hand, let\u2019s look at what would happen if you added 5 to only one side of the equation.<\/p>\n

          [latex]\\begin{array}{r}3+7=10\\\\3+7+5=10\\\\15\\neq 10\\end{array}[\/latex]<\/p>\n

          Adding 5 to only one side of the equation resulted in an equation that is false. The equation is no longer \u201cbalanced,\u201d and it is no longer a true 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

          <\/h3>\n

          When you solve an equation, you find the value of the variable that makes the equation true. In order to solve the equation, you isolate the variable<\/b>. Isolating the variable means rewriting an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.<\/p>\n

          When the equation involves addition or subtraction, use the inverse operation to \u201cundo\u201d the operation in order to isolate the variable. For addition and subtraction, your goal is to change any value being added or subtracted to 0, the additive identity.<\/p>\n

          In the following simulation, you can adjust the quantity being added or subtracted to each side of an equation to see how important it is to perform the same operation on both sides of an equation when you are solving.<\/p>\n

          \n

          Example 1<\/h3>\n

          Solve [latex]x-6=8[\/latex].<\/p>\n

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

          This equation means that if you begin with some unknown number, x<\/i>, and subtract 6, you will end up with 8. You are trying to figure out the value of the variable x.<\/i><\/p>\n

          Using the Addition Property of Equality, add 6 to both sides of the equation to isolate the variable. You choose to add 6 because 6 is being subtracted from the variable.<\/p>\n

          [latex]\\displaystyle \\begin{array}{r}x-6\\,\\,\\,=\\,\\,\\,\\,8\\\\\\,\\,\\,\\,\\,\\,\\,\\underline{+\\,6\\,\\,\\,\\,\\,\\,\\,\\,+6}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,=\\, 14\\end{array}[\/latex]<\/p>\n

          Answer<\/h4>\n

          [latex]x=14[\/latex]<\/p>\n<\/div>\n<\/div>\n

          \n

          Solve [latex]x+5=27[\/latex].<\/p>\n

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

          This equation means that if you begin with some unknown number, x<\/i>, and add 5, you will end up with 27. You are trying to figure out the value of the variable x.<\/i><\/p>\n

          Using the Addition Property of Equality, subtract 5 from both sides of the equation to isolate the variable. You choose to subtract 5, as 5 is being added from the variable.<\/p>\n

          [latex]\\displaystyle \\begin{array}{r}x+5\\,\\,=\\,\\,27\\\\\\,\\,\\,\\,\\,\\,\\,\\underline{-5\\,\\,\\,\\,\\,\\,\\,\\,-5}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,=\\, 22\\end{array}[\/latex]<\/p>\n

          Answer<\/h4>\n

          [latex]x=22[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

          In the following video two examples of using the addition property of equality are shown.
          \n