{"id":1172,"date":"2021-12-02T19:36:50","date_gmt":"2021-12-03T00:36:50","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/2-1-elementary-linear-equations\/"},"modified":"2023-08-30T12:06:04","modified_gmt":"2023-08-30T16:06:04","slug":"elementary-linear-equations","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/elementary-linear-equations\/","title":{"raw":"2.1 Elementary Linear Equations","rendered":"2.1 Elementary Linear Equations"},"content":{"raw":"Solving linear equations is an important and fundamental skill in algebra. In algebra, there are often problems in which the answer is known, but the variable part of the problem is missing. To find this missing variable, it is necessary to follow a series of steps that result in the variable equalling some solution.\r\n

Addition and Subtraction Problems<\/h1>\r\nTo solve equations, the general rule is to do the opposite of the order of operations. Consider the following.\r\n
\r\n

Example 2.1.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSolve for [latex]x.[\/latex]\r\n
    \r\n \t
  1. [latex]x-7=5[\/latex]\r\n[latex]\\begin{array}{rrrrr}\r\nx&-&7&=&-5\\\\\r\n&+&7&&+7\\\\\r\n\\hline\r\n&&x&=&2\r\n\\end{array}[\/latex]<\/li>\r\n \t
  2. [latex]4+x=8[\/latex]\r\n[latex]\\begin{array}{rrrrr}\r\n4&+&x&=&8\\\\\r\n-4&&&&-4\\\\\r\n\\hline\r\n&&x&=&4\r\n\\end{array}[\/latex]<\/li>\r\n \t
  3. [latex]7=x-9[\/latex]\r\n[latex]\\begin{array}{rrrrr}\r\n7&=&x&-&9\\\\\r\n+9&&&+&9\\\\\r\n\\hline\r\n16&=&x&&\r\n\\end{array}[\/latex]<\/li>\r\n \t
  4. [latex]5=8+x[\/latex]\r\n[latex]\\begin{array}{rrrrr}\r\n5&=&8&+&x\\\\\r\n-8&&-8&&\\\\\r\n\\hline\r\n-3&=&x&&\r\n\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

    Multiplication Problems<\/h1>\r\nIn a multiplication problem, get rid of the coefficient in front of the variable by dividing both sides of the equation by that number. Consider the following examples.\r\n
    \r\n

    Example 2.1.2<\/p>\r\n\r\n<\/header>\r\n

    \r\n\r\nSolve for [latex]x.[\/latex]\r\n
      \r\n \t
    1. [latex]\\begin{array}[t]{rrl}\r\n4x&=&20\\\\ \\\\\r\n\\dfrac {4x}{4}&=&\\dfrac{20}{4}\\\\ \\\\\r\nx&=&5\r\n\\end{array}[\/latex]<\/li>\r\n \t
    2. [latex]\\begin{array}[t]{rrl}\r\n8x&=&-24\\\\ \\\\\r\n\\dfrac {8x}{8}&=&\\dfrac{-24}{8}\\\\ \\\\\r\nx&=&-3\r\n\\end{array}[\/latex]<\/li>\r\n \t
    3. [latex]\\begin{array}[t]{rrl}\r\n-4x&=&-20\\\\ \\\\\r\n\\dfrac {-4x}{-4}&=&\\dfrac{-20}{-4}\\\\ \\\\\r\nx&=&5\r\n\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

      Division Problems<\/h1>\r\nIn division problems, remove the denominator by multiplying both sides by it. Consider the following examples.\r\n
      \r\n

      Example 2.1.3<\/p>\r\n\r\n<\/header>\r\n

      \r\n\r\nSolve for [latex]x.[\/latex]\r\n
        \r\n \t
      1. [latex]\\begin{array}[t]{rrl}\r\n\\dfrac{x}{-7}&=&-2\\\\\r\n-7\\left(\\dfrac{x}{-7}\\right)&=&(-2)-7 \\\\\r\nx&=&14\\end{array}[\/latex]<\/li>\r\n \t
      2. [latex]\\begin{array}[t]{rrl}\r\n\\dfrac{x}{8}&=&5\\\\\r\n8\\left(\\dfrac{x}{8}\\right)&=&(5)8\\\\\r\nx&=&40\\end{array}[\/latex]<\/li>\r\n \t
      3. [latex]\\begin{array}[t]{rrl}\r\n\\dfrac{x}{-4}&=&9\\\\\r\n-4\\left(\\dfrac{x}{-4}\\right)&=&(9) -4\\\\\r\nx&=&-36\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

        Questions<\/h1>\r\nFor questions 1 to 28, solve each linear equation.\r\n
          \r\n \t
        1. [latex]v + 9 = 16[\/latex]<\/li>\r\n \t
        2. [latex]14 = b + 3[\/latex]<\/li>\r\n \t
        3. [latex]x - 11 = -16[\/latex]<\/li>\r\n \t
        4. [latex]-14 = x - 18[\/latex]<\/li>\r\n \t
        5. [latex]30 = a + 20[\/latex]<\/li>\r\n \t
        6. [latex]-1 + k = 5[\/latex]<\/li>\r\n \t
        7. [latex]x - 7 = -26[\/latex]<\/li>\r\n \t
        8. [latex]-13 + p = -19[\/latex]<\/li>\r\n \t
        9. [latex]13 = n - 5[\/latex]<\/li>\r\n \t
        10. [latex]22 = 16 + m[\/latex]<\/li>\r\n \t
        11. [latex]340 = -17x[\/latex]<\/li>\r\n \t
        12. [latex]4r = -28[\/latex]<\/li>\r\n \t
        13. [latex]{-9} = \\dfrac{n}{12}[\/latex]<\/li>\r\n \t
        14. [latex]27 = 9b[\/latex]<\/li>\r\n \t
        15. [latex]20v = -160[\/latex]<\/li>\r\n \t
        16. [latex]-20x = -80[\/latex]<\/li>\r\n \t
        17. [latex]340 = 20n[\/latex]<\/li>\r\n \t
        18. [latex]12 = 8a[\/latex]<\/li>\r\n \t
        19. [latex]16x = 320[\/latex]<\/li>\r\n \t
        20. [latex]8k = -16[\/latex]<\/li>\r\n \t
        21. [latex]-16 + n = -13[\/latex]<\/li>\r\n \t
        22. [latex]-21 = x - 5[\/latex]<\/li>\r\n \t
        23. [latex]p-8 = -21[\/latex]<\/li>\r\n \t
        24. [latex]m - 4 = -13[\/latex]<\/li>\r\n \t
        25. [latex]\\dfrac{r}{14} = \\dfrac{5}{14}[\/latex]<\/li>\r\n \t
        26. [latex]\\dfrac{n}{8} = {40}[\/latex]<\/li>\r\n \t
        27. [latex]20b = -200[\/latex]<\/li>\r\n \t
        28. [latex]-\\dfrac{1}{3} = \\dfrac{x}{12}[\/latex]<\/li>\r\n<\/ol>\r\nAnswer Key 2.1<\/a>\r\n\r\n
          \r\n\r\n

          Extra Reading and Instructional Videos<\/h1>\r\nArticle to read: New theory finds 'traffic jams' in jet stream cause abnormal weather patterns<\/a>.\r\n\r\nThe abstract reads:\r\n
          A study offers an explanation for a mysterious and sometimes deadly weather pattern in which the jet stream, the global air currents that circle the Earth, stalls out over a region. Much like highways, the jet stream has a capacity, researchers said, and when it's exceeded, blockages form that are remarkably similar to traffic jams \u2014 and climate forecasters can use the same math to model them both.<\/blockquote>","rendered":"

          Solving linear equations is an important and fundamental skill in algebra. In algebra, there are often problems in which the answer is known, but the variable part of the problem is missing. To find this missing variable, it is necessary to follow a series of steps that result in the variable equalling some solution.<\/p>\n

          Addition and Subtraction Problems<\/h1>\n

          To solve equations, the general rule is to do the opposite of the order of operations. Consider the following.<\/p>\n

          \n
          \n

          Example 2.1.1<\/p>\n<\/header>\n

          \n

          Solve for [latex]x.[\/latex]<\/p>\n

            \n
          1. [latex]x-7=5[\/latex]
            \n[latex]\\begin{array}{rrrrr} x&-&7&=&-5\\\\ &+&7&&+7\\\\ \\hline &&x&=&2 \\end{array}[\/latex]<\/li>\n
          2. [latex]4+x=8[\/latex]
            \n[latex]\\begin{array}{rrrrr} 4&+&x&=&8\\\\ -4&&&&-4\\\\ \\hline &&x&=&4 \\end{array}[\/latex]<\/li>\n
          3. [latex]7=x-9[\/latex]
            \n[latex]\\begin{array}{rrrrr} 7&=&x&-&9\\\\ +9&&&+&9\\\\ \\hline 16&=&x&& \\end{array}[\/latex]<\/li>\n
          4. [latex]5=8+x[\/latex]
            \n[latex]\\begin{array}{rrrrr} 5&=&8&+&x\\\\ -8&&-8&&\\\\ \\hline -3&=&x&& \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

            Multiplication Problems<\/h1>\n

            In a multiplication problem, get rid of the coefficient in front of the variable by dividing both sides of the equation by that number. Consider the following examples.<\/p>\n

            \n
            \n

            Example 2.1.2<\/p>\n<\/header>\n

            \n

            Solve for [latex]x.[\/latex]<\/p>\n

              \n
            1. [latex]\\begin{array}[t]{rrl} 4x&=&20\\\\ \\\\ \\dfrac {4x}{4}&=&\\dfrac{20}{4}\\\\ \\\\ x&=&5 \\end{array}[\/latex]<\/li>\n
            2. [latex]\\begin{array}[t]{rrl} 8x&=&-24\\\\ \\\\ \\dfrac {8x}{8}&=&\\dfrac{-24}{8}\\\\ \\\\ x&=&-3 \\end{array}[\/latex]<\/li>\n
            3. [latex]\\begin{array}[t]{rrl} -4x&=&-20\\\\ \\\\ \\dfrac {-4x}{-4}&=&\\dfrac{-20}{-4}\\\\ \\\\ x&=&5 \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

              Division Problems<\/h1>\n

              In division problems, remove the denominator by multiplying both sides by it. Consider the following examples.<\/p>\n

              \n
              \n

              Example 2.1.3<\/p>\n<\/header>\n

              \n

              Solve for [latex]x.[\/latex]<\/p>\n

                \n
              1. [latex]\\begin{array}[t]{rrl} \\dfrac{x}{-7}&=&-2\\\\ -7\\left(\\dfrac{x}{-7}\\right)&=&(-2)-7 \\\\ x&=&14\\end{array}[\/latex]<\/li>\n
              2. [latex]\\begin{array}[t]{rrl} \\dfrac{x}{8}&=&5\\\\ 8\\left(\\dfrac{x}{8}\\right)&=&(5)8\\\\ x&=&40\\end{array}[\/latex]<\/li>\n
              3. [latex]\\begin{array}[t]{rrl} \\dfrac{x}{-4}&=&9\\\\ -4\\left(\\dfrac{x}{-4}\\right)&=&(9) -4\\\\ x&=&-36\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                Questions<\/h1>\n

                For questions 1 to 28, solve each linear equation.<\/p>\n

                  \n
                1. [latex]v + 9 = 16[\/latex]<\/li>\n
                2. [latex]14 = b + 3[\/latex]<\/li>\n
                3. [latex]x - 11 = -16[\/latex]<\/li>\n
                4. [latex]-14 = x - 18[\/latex]<\/li>\n
                5. [latex]30 = a + 20[\/latex]<\/li>\n
                6. [latex]-1 + k = 5[\/latex]<\/li>\n
                7. [latex]x - 7 = -26[\/latex]<\/li>\n
                8. [latex]-13 + p = -19[\/latex]<\/li>\n
                9. [latex]13 = n - 5[\/latex]<\/li>\n
                10. [latex]22 = 16 + m[\/latex]<\/li>\n
                11. [latex]340 = -17x[\/latex]<\/li>\n
                12. [latex]4r = -28[\/latex]<\/li>\n
                13. [latex]{-9} = \\dfrac{n}{12}[\/latex]<\/li>\n
                14. [latex]27 = 9b[\/latex]<\/li>\n
                15. [latex]20v = -160[\/latex]<\/li>\n
                16. [latex]-20x = -80[\/latex]<\/li>\n
                17. [latex]340 = 20n[\/latex]<\/li>\n
                18. [latex]12 = 8a[\/latex]<\/li>\n
                19. [latex]16x = 320[\/latex]<\/li>\n
                20. [latex]8k = -16[\/latex]<\/li>\n
                21. [latex]-16 + n = -13[\/latex]<\/li>\n
                22. [latex]-21 = x - 5[\/latex]<\/li>\n
                23. [latex]p-8 = -21[\/latex]<\/li>\n
                24. [latex]m - 4 = -13[\/latex]<\/li>\n
                25. [latex]\\dfrac{r}{14} = \\dfrac{5}{14}[\/latex]<\/li>\n
                26. [latex]\\dfrac{n}{8} = {40}[\/latex]<\/li>\n
                27. [latex]20b = -200[\/latex]<\/li>\n
                28. [latex]-\\dfrac{1}{3} = \\dfrac{x}{12}[\/latex]<\/li>\n<\/ol>\n

                  Answer Key 2.1<\/a><\/p>\n

                  \n

                  Extra Reading and Instructional Videos<\/h1>\n

                  Article to read: New theory finds ‘traffic jams’ in jet stream cause abnormal weather patterns<\/a>.<\/p>\n

                  The abstract reads:<\/p>\n

                  A study offers an explanation for a mysterious and sometimes deadly weather pattern in which the jet stream, the global air currents that circle the Earth, stalls out over a region. Much like highways, the jet stream has a capacity, researchers said, and when it’s exceeded, blockages form that are remarkably similar to traffic jams \u2014 and climate forecasters can use the same math to model them both.<\/p><\/blockquote>\n","protected":false},"author":90,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1172","chapter","type-chapter","status-publish","hentry","license-cc-by-nc-sa"],"part":1170,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1172","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":2,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1172\/revisions"}],"predecessor-version":[{"id":2066,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1172\/revisions\/2066"}],"part":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/parts\/1170"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1172\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1172"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapter-type?post=1172"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1172"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1172"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}