Solutions for Systems of Linear Equations: Three Variables

Solutions to Try Its

1. [latex]\left(1,-1,1\right)[/latex]

2. No solution.

3. Infinite number of solutions of the form [latex]\left(x,4x - 11,-5x+18\right)[/latex].

Solutions to Odd-Numbered Exercises

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

3. Not necessarily. There could be zero, one, or infinitely many solutions. For example, [latex]\left(0,0,0\right)[/latex] is not a solution to the system below, but that does not mean that it has no solution.

[latex]\begin{array}{l}\text{ }2x+3y - 6z=1\hfill \\ -4x - 6y+12z=-2\hfill \\ \text{ }x+2y+5z=10\hfill \end{array}[/latex]

5. Every system of equations can be solved graphically, by substitution, and by addition. However, systems of three equations become very complex to solve graphically so other methods are usually preferable.

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]

31. [latex]\left(-6,2,1\right)[/latex]

33. [latex]\left(5,12,15\right)[/latex]

35. [latex]\left(-5,-5,-5\right)[/latex]

37. [latex]\left(10,10,10\right)[/latex]

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

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

43. [latex]\left(2,0,0\right)[/latex]

45. [latex]\left(1,1,1\right)[/latex]

47. [latex]\left(\frac{128}{557},\frac{23}{557},\frac{28}{557}\right)[/latex]

49. [latex]\left(6,-1,0\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

61. The BMW was $49,636, the Jeep was $42,636, and the Toyota was $47,727.

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.

67. Saudi Arabia imported 16.8%, Canada imported 15.1%, and Mexico 15.0%

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