{"id":134,"date":"2026-01-12T15:50:09","date_gmt":"2026-01-12T15:50:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=134"},"modified":"2026-01-12T15:50:10","modified_gmt":"2026-01-12T15:50:10","slug":"systems-of-equations-and-inequalities-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/systems-of-equations-and-inequalities-get-stronger-answer-key\/","title":{"raw":"Systems of Equations and Inequalities: Get Stronger Answer Key","rendered":"Systems of Equations and Inequalities: Get Stronger Answer Key"},"content":{"raw":"

Systems of Linear Equations: Two Variables<\/h2>\r\n
    \r\n \t
  1. Yes<\/li>\r\n \t
  2. Yes<\/li>\r\n \t
  3. [latex](-1,2)[\/latex]<\/li>\r\n \t
  4. [latex](-3,1)[\/latex]<\/li>\r\n \t
  5. [latex](-\\dfrac{3}{5},0)[\/latex]<\/li>\r\n \t
  6. No solutions exist.<\/li>\r\n \t
  7. [latex](\\dfrac{72}{5},\\dfrac{132}{5})[\/latex]<\/li>\r\n \t
  8. [latex](6,-6)[\/latex]<\/li>\r\n \t
  9. [latex](-\\dfrac{1}{2},\\dfrac{1}{10})[\/latex]<\/li>\r\n \t
  10. No solutions exist.<\/li>\r\n \t
  11. [latex](-\\dfrac{1}{5},\\dfrac{2}{3})[\/latex]<\/li>\r\n \t
  12. [latex](x,\\dfrac{x+3}{2})[\/latex]<\/li>\r\n \t
  13. [latex](-4,4)[\/latex]<\/li>\r\n \t
  14. [latex](\\dfrac{1}{2},\\dfrac{1}{8})[\/latex]<\/li>\r\n \t
  15. [latex](\\dfrac{1}{6},0)[\/latex]<\/li>\r\n \t
  16. [latex](x,2(7x-6))[\/latex]<\/li>\r\n \t
  17. [latex](-\\dfrac{5}{6},\\dfrac{4}{3})[\/latex]<\/li>\r\n \t
  18. Consistent with one solution<\/li>\r\n \t
  19. Consistent with one solution<\/li>\r\n \t
  20. Dependent with infinitely many solutions<\/li>\r\n \t
  21. They never turn a profit.<\/li>\r\n \t
  22. [latex](1,250,100,000)[\/latex]<\/li>\r\n \t
  23. The numbers are [latex]7.5[\/latex] and [latex]20.5[\/latex].<\/li>\r\n \t
  24. [latex]24,000[\/latex]<\/li>\r\n \t
  25. [latex]790[\/latex] second-year students, [latex] 805[\/latex] first-year students<\/li>\r\n \t
  26. [latex]56[\/latex] men, [latex]74[\/latex] women<\/li>\r\n \t
  27. [latex]10[\/latex] gallons of [latex]10 \\%[\/latex] solution, [latex]15[\/latex] gallons of [latex]60 \\%[\/latex] solution<\/li>\r\n<\/ol>\r\n

    Systems of Linear Equations: Three Variables<\/h2>\r\n
      \r\n \t
    1. No<\/li>\r\n \t
    2. Yes<\/li>\r\n \t
    3. [latex](-1,4,2)[\/latex]<\/li>\r\n \t
    4. [latex](-\\dfrac{85}{107},\\dfrac{312}{107},\\dfrac{191}{107})[\/latex]<\/li>\r\n \t
    5. [latex](1,\\dfrac{1}{2},0)[\/latex]<\/li>\r\n \t
    6. [latex](4,-6,1)[\/latex]<\/li>\r\n \t
    7. [latex](x,\\dfrac{1}{27}(65-16x),\\dfrac{x+28}{27})[\/latex]<\/li>\r\n \t
    8. [latex](-\\dfrac{45}{13},\\dfrac{17}{13},-2)[\/latex]<\/li>\r\n \t
    9. No solutions exist[\/latex]<\/li>\r\n \t
    10. [latex](0,0,0)[\/latex]<\/li>\r\n \t
    11. [latex](\\dfrac{4}{7},-\\dfrac{1}{7},-\\dfrac{3}{7})[\/latex]<\/li>\r\n \t
    12. [latex]24[\/latex], [latex]36[\/latex], [latex]48[\/latex]<\/li>\r\n \t
    13. [latex]70[\/latex] grandparents, [latex]140[\/latex] parents, [latex]190[\/latex] children<\/li>\r\n \t
    14. Your share was [latex]$19.95[\/latex], Shani's share was [latex]$40[\/latex], and your other roommate's share was [latex]$22.05[\/latex].<\/li>\r\n \t
    15. There are infinitely many solutions; we need more information<\/li>\r\n \t
    16. [latex]500[\/latex] students, [latex]225[\/latex] children, and [latex]450[\/latex] adults<\/li>\r\n \t
    17. The BMW was [latex]$49,636[\/latex], the Jeep was [latex]$42,636[\/latex], and the Toyota was [latex]$47,727[\/latex].<\/li>\r\n<\/ol>\r\n

      Systems of Nonlinear Equations and Inequalities<\/h2>\r\n
        \r\n \t
      1. [latex](0,-3), (3,0)[\/latex]<\/li>\r\n \t
      2. [latex](-\\dfrac{3\\sqrt{2}}{2},\\dfrac{3\\sqrt{2}}{2}), (\\dfrac{3\\sqrt{2}}{2},-\\dfrac{3\\sqrt{2}}{2})[\/latex]<\/li>\r\n \t
      3. [latex](-3,0), (3,0)[\/latex]<\/li>\r\n \t
      4. [latex](\\dfrac{1}{4},-\\dfrac{\\sqrt{62}}{8}), (\\dfrac{1}{4},\\dfrac{\\sqrt{62}}{8})[\/latex]<\/li>\r\n \t
      5. [latex](-\\dfrac{\\sqrt{398}}{4},-\\dfrac{199}{4}), (\\dfrac{\\sqrt{398}}{4},-\\dfrac{199}{4})[\/latex]<\/li>\r\n \t
      6. [latex](0,2), (1,3)[\/latex]<\/li>\r\n \t
      7. [latex](-\\sqrt{\\dfrac{1}{2}}(\\sqrt{5}-1),\\dfrac{1}{2}(1-\\sqrt{5})), (\\sqrt{\\dfrac{1}{2}}(\\sqrt{5}-1),\\dfrac{1}{2}(1-\\sqrt{5}))[\/latex]<\/li>\r\n \t
      8. [latex](5,0)[\/latex]<\/li>\r\n \t
      9. [latex](0,0)[\/latex]<\/li>\r\n \t
      10. [latex](3,0)[\/latex]<\/li>\r\n \t
      11. No Solutions Exist<\/li>\r\n \t
      12. No Solutions Exist<\/li>\r\n \t
      13. [latex](-\\dfrac{\\sqrt{2}}{2},-\\dfrac{\\sqrt{2}}{2}), (-\\dfrac{\\sqrt{2}}{2},\\dfrac{\\sqrt{2}}{2}), (\\dfrac{\\sqrt{2}}{2},-\\dfrac{\\sqrt{2}}{2}), (\\dfrac{\\sqrt{2}}{2},\\dfrac{\\sqrt{2}}{2})[\/latex]<\/li>\r\n \t
      14. [latex](2,0)[\/latex]<\/li>\r\n \t
      15. \"Graph<\/li>\r\n \t
      16. \"Graph<\/li>\r\n \t
      17. \"<\/li>\r\n \t
      18. \"<\/li>\r\n \t
      19. [latex]12, 288[\/latex]<\/li>\r\n \t
      20. [latex]2\u201320[\/latex] computers<\/li>\r\n<\/ol>\r\n

        Partial Fraction Decomposition<\/h2>\r\n
          \r\n \t
        1. [latex]\\dfrac{8}{x+3}-\\dfrac{5}{x-8}[\/latex]<\/li>\r\n \t
        2. [latex]\\dfrac{1}{x+5}+\\dfrac{9}{x+2}[\/latex]<\/li>\r\n \t
        3. [latex]\\dfrac{3}{5x-2}+\\dfrac{4}{4x-1}[\/latex]<\/li>\r\n \t
        4. [latex]\\dfrac{5}{2(x+3)}+\\dfrac{5}{2(x-3)}[\/latex]<\/li>\r\n \t
        5. [latex]\\dfrac{1}{x-2}+\\dfrac{2}{(x-2)^2}[\/latex]<\/li>\r\n \t
        6. [latex]-\\dfrac{6}{4x+5}+\\dfrac{3}{(4x+5)^2}[\/latex]<\/li>\r\n \t
        7. [latex]-\\dfrac{1}{x-7}+\\dfrac{2}{(x-7)^2}[\/latex]<\/li>\r\n \t
        8. [latex]\\dfrac{4}{x}-\\dfrac{3}{2(x+1)}+\\dfrac{7}{2(x+1)^2}[\/latex]<\/li>\r\n \t
        9. [latex]\\dfrac{x+1}{x^2+x+3}+\\dfrac{3}{x+2}[\/latex]<\/li>\r\n \t
        10. [latex]\\dfrac{4-3x}{x^2+3x+8}+\\dfrac{1}{x-1}[\/latex]<\/li>\r\n \t
        11. [latex]\\dfrac{2x-1}{x^2+6x+1}+\\dfrac{2}{x+3}[\/latex]<\/li>\r\n \t
        12. [latex]\\dfrac{1}{x^2+x+1}+\\dfrac{4}{x-1}[\/latex]<\/li>\r\n \t
        13. [latex]\\dfrac{x+6}{x^2+1}+\\dfrac{4x+3}{(x^2+1)^2}[\/latex]<\/li>\r\n \t
        14. [latex]\\dfrac{x+1}{x+2}+\\dfrac{2x+3}{(x+2)^2}[\/latex]<\/li>\r\n \t
        15. [latex]\\dfrac{1}{x^2+3x+25}+\\dfrac{3x}{(x^2+3x+25)^2}[\/latex]<\/li>\r\n \t
        16. [latex]\\dfrac{1}{8x}-\\dfrac{x}{8(x^2+4)}+\\dfrac{10-x}{2(x^2+4)^2}[\/latex]<\/li>\r\n<\/ol>","rendered":"

          Systems of Linear Equations: Two Variables<\/h2>\n
            \n
          1. Yes<\/li>\n
          2. Yes<\/li>\n
          3. [latex](-1,2)[\/latex]<\/li>\n
          4. [latex](-3,1)[\/latex]<\/li>\n
          5. [latex](-\\dfrac{3}{5},0)[\/latex]<\/li>\n
          6. No solutions exist.<\/li>\n
          7. [latex](\\dfrac{72}{5},\\dfrac{132}{5})[\/latex]<\/li>\n
          8. [latex](6,-6)[\/latex]<\/li>\n
          9. [latex](-\\dfrac{1}{2},\\dfrac{1}{10})[\/latex]<\/li>\n
          10. No solutions exist.<\/li>\n
          11. [latex](-\\dfrac{1}{5},\\dfrac{2}{3})[\/latex]<\/li>\n
          12. [latex](x,\\dfrac{x+3}{2})[\/latex]<\/li>\n
          13. [latex](-4,4)[\/latex]<\/li>\n
          14. [latex](\\dfrac{1}{2},\\dfrac{1}{8})[\/latex]<\/li>\n
          15. [latex](\\dfrac{1}{6},0)[\/latex]<\/li>\n
          16. [latex](x,2(7x-6))[\/latex]<\/li>\n
          17. [latex](-\\dfrac{5}{6},\\dfrac{4}{3})[\/latex]<\/li>\n
          18. Consistent with one solution<\/li>\n
          19. Consistent with one solution<\/li>\n
          20. Dependent with infinitely many solutions<\/li>\n
          21. They never turn a profit.<\/li>\n
          22. [latex](1,250,100,000)[\/latex]<\/li>\n
          23. The numbers are [latex]7.5[\/latex] and [latex]20.5[\/latex].<\/li>\n
          24. [latex]24,000[\/latex]<\/li>\n
          25. [latex]790[\/latex] second-year students, [latex]805[\/latex] first-year students<\/li>\n
          26. [latex]56[\/latex] men, [latex]74[\/latex] women<\/li>\n
          27. [latex]10[\/latex] gallons of [latex]10 \\%[\/latex] solution, [latex]15[\/latex] gallons of [latex]60 \\%[\/latex] solution<\/li>\n<\/ol>\n

            Systems of Linear Equations: Three Variables<\/h2>\n
              \n
            1. No<\/li>\n
            2. Yes<\/li>\n
            3. [latex](-1,4,2)[\/latex]<\/li>\n
            4. [latex](-\\dfrac{85}{107},\\dfrac{312}{107},\\dfrac{191}{107})[\/latex]<\/li>\n
            5. [latex](1,\\dfrac{1}{2},0)[\/latex]<\/li>\n
            6. [latex](4,-6,1)[\/latex]<\/li>\n
            7. [latex](x,\\dfrac{1}{27}(65-16x),\\dfrac{x+28}{27})[\/latex]<\/li>\n
            8. [latex](-\\dfrac{45}{13},\\dfrac{17}{13},-2)[\/latex]<\/li>\n
            9. No solutions exist[\/latex]<\/li>\n
            10. [latex](0,0,0)[\/latex]<\/li>\n
            11. [latex](\\dfrac{4}{7},-\\dfrac{1}{7},-\\dfrac{3}{7})[\/latex]<\/li>\n
            12. [latex]24[\/latex], [latex]36[\/latex], [latex]48[\/latex]<\/li>\n
            13. [latex]70[\/latex] grandparents, [latex]140[\/latex] parents, [latex]190[\/latex] children<\/li>\n
            14. Your share was [latex]$19.95[\/latex], Shani’s share was [latex]$40[\/latex], and your other roommate’s share was [latex]$22.05[\/latex].<\/li>\n
            15. There are infinitely many solutions; we need more information<\/li>\n
            16. [latex]500[\/latex] students, [latex]225[\/latex] children, and [latex]450[\/latex] adults<\/li>\n
            17. The BMW was [latex]$49,636[\/latex], the Jeep was [latex]$42,636[\/latex], and the Toyota was [latex]$47,727[\/latex].<\/li>\n<\/ol>\n

              Systems of Nonlinear Equations and Inequalities<\/h2>\n
                \n
              1. [latex](0,-3), (3,0)[\/latex]<\/li>\n
              2. [latex](-\\dfrac{3\\sqrt{2}}{2},\\dfrac{3\\sqrt{2}}{2}), (\\dfrac{3\\sqrt{2}}{2},-\\dfrac{3\\sqrt{2}}{2})[\/latex]<\/li>\n
              3. [latex](-3,0), (3,0)[\/latex]<\/li>\n
              4. [latex](\\dfrac{1}{4},-\\dfrac{\\sqrt{62}}{8}), (\\dfrac{1}{4},\\dfrac{\\sqrt{62}}{8})[\/latex]<\/li>\n
              5. [latex](-\\dfrac{\\sqrt{398}}{4},-\\dfrac{199}{4}), (\\dfrac{\\sqrt{398}}{4},-\\dfrac{199}{4})[\/latex]<\/li>\n
              6. [latex](0,2), (1,3)[\/latex]<\/li>\n
              7. [latex](-\\sqrt{\\dfrac{1}{2}}(\\sqrt{5}-1),\\dfrac{1}{2}(1-\\sqrt{5})), (\\sqrt{\\dfrac{1}{2}}(\\sqrt{5}-1),\\dfrac{1}{2}(1-\\sqrt{5}))[\/latex]<\/li>\n
              8. [latex](5,0)[\/latex]<\/li>\n
              9. [latex](0,0)[\/latex]<\/li>\n
              10. [latex](3,0)[\/latex]<\/li>\n
              11. No Solutions Exist<\/li>\n
              12. No Solutions Exist<\/li>\n
              13. [latex](-\\dfrac{\\sqrt{2}}{2},-\\dfrac{\\sqrt{2}}{2}), (-\\dfrac{\\sqrt{2}}{2},\\dfrac{\\sqrt{2}}{2}), (\\dfrac{\\sqrt{2}}{2},-\\dfrac{\\sqrt{2}}{2}), (\\dfrac{\\sqrt{2}}{2},\\dfrac{\\sqrt{2}}{2})[\/latex]<\/li>\n
              14. [latex](2,0)[\/latex]<\/li>\n
              15. \"Graph<\/li>\n
              16. \"Graph<\/li>\n
              17. \"Graph<\/li>\n
              18. \"Graph<\/li>\n
              19. [latex]12, 288[\/latex]<\/li>\n
              20. [latex]2\u201320[\/latex] computers<\/li>\n<\/ol>\n

                Partial Fraction Decomposition<\/h2>\n
                  \n
                1. [latex]\\dfrac{8}{x+3}-\\dfrac{5}{x-8}[\/latex]<\/li>\n
                2. [latex]\\dfrac{1}{x+5}+\\dfrac{9}{x+2}[\/latex]<\/li>\n
                3. [latex]\\dfrac{3}{5x-2}+\\dfrac{4}{4x-1}[\/latex]<\/li>\n
                4. [latex]\\dfrac{5}{2(x+3)}+\\dfrac{5}{2(x-3)}[\/latex]<\/li>\n
                5. [latex]\\dfrac{1}{x-2}+\\dfrac{2}{(x-2)^2}[\/latex]<\/li>\n
                6. [latex]-\\dfrac{6}{4x+5}+\\dfrac{3}{(4x+5)^2}[\/latex]<\/li>\n
                7. [latex]-\\dfrac{1}{x-7}+\\dfrac{2}{(x-7)^2}[\/latex]<\/li>\n
                8. [latex]\\dfrac{4}{x}-\\dfrac{3}{2(x+1)}+\\dfrac{7}{2(x+1)^2}[\/latex]<\/li>\n
                9. [latex]\\dfrac{x+1}{x^2+x+3}+\\dfrac{3}{x+2}[\/latex]<\/li>\n
                10. [latex]\\dfrac{4-3x}{x^2+3x+8}+\\dfrac{1}{x-1}[\/latex]<\/li>\n
                11. [latex]\\dfrac{2x-1}{x^2+6x+1}+\\dfrac{2}{x+3}[\/latex]<\/li>\n
                12. [latex]\\dfrac{1}{x^2+x+1}+\\dfrac{4}{x-1}[\/latex]<\/li>\n
                13. [latex]\\dfrac{x+6}{x^2+1}+\\dfrac{4x+3}{(x^2+1)^2}[\/latex]<\/li>\n
                14. [latex]\\dfrac{x+1}{x+2}+\\dfrac{2x+3}{(x+2)^2}[\/latex]<\/li>\n
                15. [latex]\\dfrac{1}{x^2+3x+25}+\\dfrac{3x}{(x^2+3x+25)^2}[\/latex]<\/li>\n
                16. [latex]\\dfrac{1}{8x}-\\dfrac{x}{8(x^2+4)}+\\dfrac{10-x}{2(x^2+4)^2}[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":13,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":106,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/134"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/134\/revisions"}],"predecessor-version":[{"id":142,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/134\/revisions\/142"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/106"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/134\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=134"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=134"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=134"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}