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]