Systems of Linear Equations: Two Variables

1. No, you can either have zero, one, or infinitely many. Examine graphs.

5. You can solve by substitution (isolating [latex]x[/latex] or [latex]y[/latex] ), graphically, or by addition.

7. Yes

9. Yes

11. [latex]\left(-1,2\right)[/latex]

13. [latex]\left(-3,1\right)[/latex]

15. [latex]\left(-\frac{3}{5},0\right)[/latex]

17. No solutions exist.

19. [latex]\left(\frac{72}{5},\frac{132}{5}\right)[/latex]

21. [latex]\left(6,-6\right)[/latex]

23. [latex]\left(-\frac{1}{2},\frac{1}{10}\right)[/latex]

25. No solutions exist.

27. [latex]\left(-\frac{1}{5},\frac{2}{3}\right)[/latex]

29. [latex]\left(x,\frac{x+3}{2}\right)[/latex]

31. [latex]\left(-4,4\right)[/latex]

33. [latex]\left(\frac{1}{2},\frac{1}{8}\right)[/latex]

35. [latex]\left(\frac{1}{6},0\right)[/latex]

37. [latex]\left(x,2\left(7x - 6\right)\right)[/latex]

39. [latex]\left(-\frac{5}{6},\frac{4}{3}\right)[/latex]

41. Consistent with one solution

43. Consistent with one solution

45. Dependent with infinitely many solutions

47. [latex]\left(-3.08,4.91\right)[/latex]

49. [latex]\left(-1.52,2.29\right)[/latex]

61. The numbers are 7.5 and 20.5.

63. 24,000

65. 790 sophomores, 805 freshman

67. 56 men, 74 women

69. 10 gallons of 10% solution, 15 gallons of 60% solution

73. $12,500 in the first account, $10,500 in the second account.

Systems of Linear Equations: Three Variables

1. No, there can be only one, zero, or infinitely many solutions.

7. No

9. Yes

11. [latex]\left(-1,4,2\right)[/latex]

13. [latex]\left(-\frac{85}{107},\frac{312}{107},\frac{191}{107}\right)[/latex]

15. [latex]\left(1,\frac{1}{2},0\right)[/latex]

17. [latex]\left(4,-6,1\right)[/latex]

19. [latex]\left(x,\frac{1}{27}\left(65 - 16x\right),\frac{x+28}{27}\right)[/latex]

21. [latex]\left(-\frac{45}{13},\frac{17}{13},-2\right)[/latex]

23. No solutions exist

25. [latex]\left(0,0,0\right)[/latex]

27. [latex]\left(\frac{4}{7},-\frac{1}{7},-\frac{3}{7}\right)[/latex]

29. [latex]\left(7,20,16\right)[/latex]

39. [latex]\left(\frac{1}{2},\frac{1}{5},\frac{4}{5}\right)[/latex]

43. [latex]\left(2,0,0\right)[/latex]

51. 24, 36, 48

53. 70 grandparents, 140 parents, 190 children

55. Your share was $19.95, Sarah’s share was $40, and your other roommate’s share was $22.05.

57. There are infinitely many solutions; we need more information

59. 500 students, 225 children, and 450 adults

63. $400,000 in the account that pays 3% interest, $500,000 in the account that pays 4% interest, and $100,000 in the account that pays 2% interest.

65. The United States consumed 26.3%, Japan 7.1%, and China 6.4% of the world’s oil.

69. Birds were 19.3%, fish were 18.6%, and mammals were 17.1% of endangered species

Systems of Nonlinear Equations and Inequalities: Two Variables

Solutions to Odd-Numbered Exercises

3. No. There does not need to be a feasible region. Consider a system that is bounded by two parallel lines. One inequality represents the region above the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region.

7. [latex]\left(0,-3\right),\left(3,0\right)[/latex]

9. [latex]\left(-\frac{3\sqrt{2}}{2},\frac{3\sqrt{2}}{2}\right),\left(\frac{3\sqrt{2}}{2},-\frac{3\sqrt{2}}{2}\right)[/latex]

11. [latex]\left(-3,0\right),\left(3,0\right)[/latex]

13. [latex]\left(\frac{1}{4},-\frac{\sqrt{62}}{8}\right),\left(\frac{1}{4},\frac{\sqrt{62}}{8}\right)[/latex]

15. [latex]\left(-\frac{\sqrt{398}}{4},\frac{199}{4}\right),\left(\frac{\sqrt{398}}{4},\frac{199}{4}\right)[/latex]

17. [latex]\left(0,2\right),\left(1,3\right)[/latex]

19. [latex]\left(-\sqrt{\frac{1}{2}\left(\sqrt{5}-1\right)},\frac{1}{2}\left(1-\sqrt{5}\right)\right),\left(\sqrt{\frac{1}{2}\left(\sqrt{5}-1\right)},\frac{1}{2}\left(1-\sqrt{5}\right)\right)[/latex]

21. [latex]\left(5,0\right)[/latex]

25. [latex]\left(3,0\right)[/latex]

27. No Solutions Exist

33. [latex]\left(2,0\right)[/latex]

35. [latex]\left(-\sqrt{7},-3\right),\left(-\sqrt{7},3\right),\left(\sqrt{7},-3\right),\left(\sqrt{7},3\right)[/latex]

41.

43.

45.

49. [latex]\left(-2\sqrt{\frac{70}{383}},-2\sqrt{\frac{35}{29}}\right),\left(-2\sqrt{\frac{70}{383}},2\sqrt{\frac{35}{29}}\right),\left(2\sqrt{\frac{70}{383}},-2\sqrt{\frac{35}{29}}\right),\left(2\sqrt{\frac{70}{383}},2\sqrt{\frac{35}{29}}\right)[/latex]

51. No Solution Exists

55. 12, 288

57. 2–20 computers

Partial Fractions

Solutions to Odd-Numbered Exercises

4. Find a common denominator and condense your partial fractions back into a single fraction. If it matches the original expression it was decomposed correctly.

7. [latex]\frac{8}{x+3}-\frac{5}{x - 8}[/latex]

11. [latex]\frac{3}{5x - 2}+\frac{4}{4x - 1}[/latex]

13. [latex]\frac{5}{2\left(x+3\right)}+\frac{5}{2\left(x - 3\right)}[/latex]

17. [latex]\frac{9}{5\left(x+2\right)}+\frac{11}{5\left(x - 3\right)}[/latex]

21. [latex]\frac{1}{x - 2}+\frac{2}{{\left(x - 2\right)}^{2}}[/latex]

23. [latex]-\frac{6}{4x+5}+\frac{3}{{\left(4x+5\right)}^{2}}[/latex]

27. [latex]\frac{4}{x}-\frac{3}{2\left(x+1\right)}+\frac{7}{2{\left(x+1\right)}^{2}}[/latex]

29. [latex]\frac{4}{x}+\frac{2}{{x}^{2}}-\frac{3}{3x+2}+\frac{7}{2{\left(3x+2\right)}^{2}}[/latex]

31. [latex]\frac{x+1}{{x}^{2}+x+3}+\frac{3}{x+2}[/latex]

33. [latex]\frac{4 - 3x}{{x}^{2}+3x+8}+\frac{1}{x - 1}[/latex]

37. [latex]\frac{1}{{x}^{2}+x+1}+\frac{4}{x - 1}[/latex]

41. [latex]-\frac{1}{4{x}^{2}+6x+9}+\frac{1}{2x - 3}[/latex]

43. [latex]\frac{1}{x}+\frac{1}{x+6}-\frac{4x}{{x}^{2}-6x+36}[/latex]

45. [latex]\frac{x+6}{{x}^{2}+1}+\frac{4x+3}{{\left({x}^{2}+1\right)}^{2}}[/latex]

47. [latex]\frac{x+1}{x+2}+\frac{2x+3}{{\left(x+2\right)}^{2}}[/latex]

49. [latex]\frac{1}{{x}^{2}+3x+25}-\frac{3x}{{\left({x}^{2}+3x+25\right)}^{2}}[/latex]

51. [latex]\frac{1}{8x}-\frac{x}{8\left({x}^{2}+4\right)}+\frac{10-x}{2{\left({x}^{2}+4\right)}^{2}}[/latex]

53. [latex]-\frac{16}{x}-\frac{9}{{x}^{2}}+\frac{16}{x - 1}-\frac{7}{{\left(x - 1\right)}^{2}}[/latex]

57. [latex]\frac{5}{x - 2}-\frac{3}{10\left(x+2\right)}+\frac{7}{x+8}-\frac{7}{10\left(x - 8\right)}[/latex]

59. [latex]-\frac{5}{4x}-\frac{5}{2\left(x+2\right)}+\frac{11}{2\left(x+4\right)}+\frac{5}{4\left(x+4\right)}[/latex]