Systems of Equations and Inequalities: Get Stronger Answer Key

Systems of Linear Equations: Two Variables

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

Systems of Linear Equations: Three Variables

  1. No
  2. Yes
  3. [latex](-1,4,2)[/latex]
  4. [latex](-\dfrac{85}{107},\dfrac{312}{107},\dfrac{191}{107})[/latex]
  5. [latex](1,\dfrac{1}{2},0)[/latex]
  6. [latex](4,-6,1)[/latex]
  7. [latex](x,\dfrac{1}{27}(65-16x),\dfrac{x+28}{27})[/latex]
  8. [latex](-\dfrac{45}{13},\dfrac{17}{13},-2)[/latex]
  9. No solutions exist[/latex]
  10. [latex](0,0,0)[/latex]
  11. [latex](\dfrac{4}{7},-\dfrac{1}{7},-\dfrac{3}{7})[/latex]
  12. [latex]24[/latex], [latex]36[/latex], [latex]48[/latex]
  13. [latex]70[/latex] grandparents, [latex]140[/latex] parents, [latex]190[/latex] children
  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].
  15. There are infinitely many solutions; we need more information
  16. [latex]500[/latex] students, [latex]225[/latex] children, and [latex]450[/latex] adults
  17. The BMW was [latex]$49,636[/latex], the Jeep was [latex]$42,636[/latex], and the Toyota was [latex]$47,727[/latex].

Systems of Nonlinear Equations and Inequalities

  1. [latex](0,-3), (3,0)[/latex]
  2. [latex](-\dfrac{3\sqrt{2}}{2},\dfrac{3\sqrt{2}}{2}), (\dfrac{3\sqrt{2}}{2},-\dfrac{3\sqrt{2}}{2})[/latex]
  3. [latex](-3,0), (3,0)[/latex]
  4. [latex](\dfrac{1}{4},-\dfrac{\sqrt{62}}{8}), (\dfrac{1}{4},\dfrac{\sqrt{62}}{8})[/latex]
  5. [latex](-\dfrac{\sqrt{398}}{4},-\dfrac{199}{4}), (\dfrac{\sqrt{398}}{4},-\dfrac{199}{4})[/latex]
  6. [latex](0,2), (1,3)[/latex]
  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]
  8. [latex](5,0)[/latex]
  9. [latex](0,0)[/latex]
  10. [latex](3,0)[/latex]
  11. No Solutions Exist
  12. No Solutions Exist
  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]
  14. [latex](2,0)[/latex]
  15. Graph of an upward-facing parabola with a shaded region inside, extending from approximately -5 to 5 on the x-axis and -10 to 10 on the y-axis. The parabola's outline is represented by a dashed curve, and arrows indicate the curve’s upward direction.
  16. Graph of a shaded region bounded by a curved, dashed line extending from the origin (0, 0) outward. Key points are labeled at approximately (√2 - 1, 2(√2 - 1)) near the top-right of the shaded area and (-1 - √2, -2(1 + √2)) near the bottom-left. The graph includes grid lines and labeled axes.
  17. Graph showing two symmetric shaded regions along the y-axis, each bounded by a dashed curve. The key points are labeled at approximately (-√37/2, 3√7/2), (√37/2, 3√7/2) above the x-axis, and (-√37/2, -3√7/2), (√37/2, -3√7/2) below the x-axis. The graph includes grid lines and labeled axes.
  18. Graph showing two symmetric shaded regions along the y-axis, each bounded by dashed, outward-opening lines. Key points are labeled at approximately (-√19/10, √47/10), (√19/10, √47/10) at the top and (-√19/10, -√47/10), (√19/10, -√47/10) at the bottom. The graph includes grid lines, labeled axes, and arrows indicating the direction of the curves.
  19. [latex]12, 288[/latex]
  20. [latex]2–20[/latex] computers

Partial Fraction Decomposition

  1. [latex]\dfrac{8}{x+3}-\dfrac{5}{x-8}[/latex]
  2. [latex]\dfrac{1}{x+5}+\dfrac{9}{x+2}[/latex]
  3. [latex]\dfrac{3}{5x-2}+\dfrac{4}{4x-1}[/latex]
  4. [latex]\dfrac{5}{2(x+3)}+\dfrac{5}{2(x-3)}[/latex]
  5. [latex]\dfrac{1}{x-2}+\dfrac{2}{(x-2)^2}[/latex]
  6. [latex]-\dfrac{6}{4x+5}+\dfrac{3}{(4x+5)^2}[/latex]
  7. [latex]-\dfrac{1}{x-7}+\dfrac{2}{(x-7)^2}[/latex]
  8. [latex]\dfrac{4}{x}-\dfrac{3}{2(x+1)}+\dfrac{7}{2(x+1)^2}[/latex]
  9. [latex]\dfrac{x+1}{x^2+x+3}+\dfrac{3}{x+2}[/latex]
  10. [latex]\dfrac{4-3x}{x^2+3x+8}+\dfrac{1}{x-1}[/latex]
  11. [latex]\dfrac{2x-1}{x^2+6x+1}+\dfrac{2}{x+3}[/latex]
  12. [latex]\dfrac{1}{x^2+x+1}+\dfrac{4}{x-1}[/latex]
  13. [latex]\dfrac{x+6}{x^2+1}+\dfrac{4x+3}{(x^2+1)^2}[/latex]
  14. [latex]\dfrac{x+1}{x+2}+\dfrac{2x+3}{(x+2)^2}[/latex]
  15. [latex]\dfrac{1}{x^2+3x+25}+\dfrac{3x}{(x^2+3x+25)^2}[/latex]
  16. [latex]\dfrac{1}{8x}-\dfrac{x}{8(x^2+4)}+\dfrac{10-x}{2(x^2+4)^2}[/latex]