1. Can a linear system of three equations have exactly two solutions? Explain why or why not

2. If a given ordered triple solves the system of equations, is that solution unique? If so, explain why. If not, give an example where it is not unique.

3. If a given ordered triple does not solve the system of equations, is there no solution? If so, explain why. If not, give an example.

4. Using the method of addition, is there only one way to solve the system?

5. Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.

6. [latex]\begin{align}2x - 6y+6z&=-12\\x+4y+5z&=-1 \\ -x+2y+3z&=-1 \end{align}[/latex] and [latex]\left(0,1,-1\right)[/latex]

7. [latex]\begin{align}6x-y+3z&=6 \\ 3x+5y+2z&=0 \\ x+y&=0 \end{align}[/latex] and [latex]\left(3,-3,-5\right)[/latex]

8. [latex]\begin{align}6x - 7y+z&=2 \\ -x-y+3z&=4 \\ 2x+y-z&=1 \end{align}[/latex] and [latex]\left(4,2,-6\right)[/latex]

9. [latex]\begin{align}x-y&=0 \\ x-z&=5 \\ x-y+z&=-1 \end{align}[/latex] and [latex]\left(4,4,-1\right)[/latex]

10. [latex]\begin{align} -x-y+2z&=3 \\ 5x+8y - 3z&=4 \\ -x+3y - 5z&=-5 \end{align}[/latex] and [latex]\left(4,1,-7\right)[/latex]

For the following exercises, solve each system by substitution.

11. [latex]\begin{align}3x - 4y+2z&=-15 \\ 2x+4y+z&=16 \\ 2x+3y+5z&=20 \end{align}[/latex]

12. [latex]\begin{align}5x - 2y+3z&=20 \\ 2x - 4y - 3z&=-9 \\ x+6y - 8z&=21 \end{align}[/latex]

13. [latex]\begin{align}5x+2y+4z&=9 \\ -3x+2y+z&=10 \\ 4x - 3y+5z&=-3 \end{align}[/latex]

14. [latex]\begin{align}4x - 3y+5z&=31 \\ -x+2y+4z&=20 \\ x+5y - 2z&=-29 \end{align}[/latex]

15. [latex]\begin{align}5x - 2y+3z&=4 \\ -4x+6y - 7z&=-1 \\ 3x+2y-z&=4\end{align}[/latex]

16. [latex]\begin{align} 4x+6y+9z&=0 \\ -5x+2y - 6z&=3 \\ 7x - 4y+3z&=-3 \end{align}[/latex]

For the following exercises, solve each system by Gaussian elimination.

17. [latex]\begin{align}2x-y+3z&=17 \\ -5x+4y - 2z&=-46 \\ 2y+5z&=-7 \end{align}[/latex]

18. [latex]\begin{align}5x - 6y+3z&=50 \\ -x+4y&=10 \\ 2x-z&=10 \end{align}[/latex]

19. [latex]\begin{align}2x+3y - 6z&=1 \\ -4x - 6y+12z&=-2 \\ x+2y+5z&=10 \end{align}[/latex]

20. [latex]\begin{align}4x+6y - 2z&=8 \\ 6x+9y - 3z&=12 \\ -2x - 3y+z&=-4 \end{align}[/latex]

21. [latex]\begin{align}2x+3y - 4z&=5 \\ -3x+2y+z&=11 \\ -x+5y+3z&=4 \end{align}[/latex]

22. [latex]\begin{align}10x+2y - 14z&=8 \\ -x-2y - 4z&=-1 \\ -12x - 6y+6z&=-12 \end{align}[/latex]

23. [latex]\begin{align}x+y+z&=14 \\ 2y+3z&=-14 \\ -16y - 24z&=-112 \end{align}[/latex]

24. [latex]\begin{align}5x - 3y+4z&=-1 \\ -4x+2y - 3z&=0 \\ -x+5y+7z&=-11 \end{align}[/latex]

25. [latex]\begin{align}x+y+z&=0 \\ 2x-y+3z&=0 \\ x-z&=0 \end{align}[/latex]

26. [latex]\begin{align}3x+2y - 5z&=6\\ 5x - 4y+3z&=-12\\ 4x+5y - 2z&=15\end{align}[/latex]

27. [latex]\begin{align}x+y+z&=0\\ 2x-y+3z&=0 \\ x-z&=1 \end{align}[/latex]

28. [latex]\begin{align} 3x-\frac{1}{2}y-z&=-\frac{1}{2} \\ 4x+z&=3 \\ -x+\frac{3}{2}y&=\frac{5}{2} \end{align}[/latex]

29. [latex]\begin{align}6x - 5y+6z&=38 \\ \frac{1}{5}x-\frac{1}{2}y+\frac{3}{5}z&=1 \\ -4x-\frac{3}{2}y-z&=-74 \end{align}[/latex]

30. [latex]\begin{align}\frac{1}{2}x-\frac{1}{5}y+\frac{2}{5}z&=-\frac{13}{10} \\ \frac{1}{4}x-\frac{2}{5}y-\frac{1}{5}z&=-\frac{7}{20} \\ -\frac{1}{2}x-\frac{3}{4}y-\frac{1}{2}z&=-\frac{5}{4} \end{align}[/latex]

31. [latex]\begin{align} -\frac{1}{3}x-\frac{1}{2}y-\frac{1}{4}z&=\frac{3}{4} \\ -\frac{1}{2}x-\frac{1}{4}y-\frac{1}{2}z&=2 \\ -\frac{1}{4}x-\frac{3}{4}y-\frac{1}{2}z&=-\frac{1}{2} \end{align}[/latex]

32. [latex]\begin{align}\frac{1}{2}x-\frac{1}{4}y+\frac{3}{4}z&=0\\ \frac{1}{4}x-\frac{1}{10}y+\frac{2}{5}z&=-2\\ \frac{1}{8}x+\frac{1}{5}y-\frac{1}{8}z&=2\end{align}[/latex]

33. [latex]\begin{align}\frac{4}{5}x-\frac{7}{8}y+\frac{1}{2}z&=1 \\ -\frac{4}{5}x-\frac{3}{4}y+\frac{1}{3}z&=-8 \\ -\frac{2}{5}x-\frac{7}{8}y+\frac{1}{2}z&=-5 \end{align}[/latex]

34. [latex]\begin{align} -\frac{1}{3}x-\frac{1}{8}y+\frac{1}{6}z&=-\frac{4}{3} \\ -\frac{2}{3}x-\frac{7}{8}y+\frac{1}{3}z&=-\frac{23}{3} \\ -\frac{1}{3}x-\frac{5}{8}y+\frac{5}{6}z&=0 \end{align}[/latex]

35. [latex]\begin{align} -\frac{1}{4}x-\frac{5}{4}y+\frac{5}{2}z&=-5 \\ -\frac{1}{2}x-\frac{5}{3}y+\frac{5}{4}z&=\frac{55}{12} \\ -\frac{1}{3}x-\frac{1}{3}y+\frac{1}{3}z&=\frac{5}{3} \end{align}[/latex]

36. [latex]\begin{align}\frac{1}{40}x+\frac{1}{60}y+\frac{1}{80}z&=\frac{1}{100} \\ -\frac{1}{2}x-\frac{1}{3}y-\frac{1}{4}z&=-\frac{1}{5} \\ \frac{3}{8}x+\frac{3}{12}y+\frac{3}{16}z&=\frac{3}{20} \end{align}[/latex]

37. [latex]\begin{align}0.1x - 0.2y+0.3z&=2\\ 0.5x - 0.1y+0.4z&=8\\ 0.7x - 0.2y+0.3z&=8\end{align}[/latex]

38. [latex]\begin{align}0.2x+0.1y - 0.3z&=0.2\\ 0.8x+0.4y - 1.2z&=0.1\\ 1.6x+0.8y - 2.4z&=0.2\end{align}[/latex]

39. [latex]\begin{align}1.1x+0.7y - 3.1z&=-1.79\\ 2.1x+0.5y - 1.6z&=-0.13\\ 0.5x+0.4y - 0.5z&=-0.07\end{align}[/latex]

40. [latex]\begin{align}0.5x - 0.5y+0.5z&=10\\ 0.2x - 0.2y+0.2z&=4\\ 0.1x - 0.1y+0.1z&=2\end{align}[/latex]

41. [latex]\begin{align}0.1x+0.2y+0.3z&=0.37\\ 0.1x - 0.2y - 0.3z&=-0.27\\ 0.5x - 0.1y - 0.3z&=-0.03\end{align}[/latex]

42. [latex]\begin{align}0.5x - 0.5y - 0.3z&=0.13\\ 0.4x - 0.1y - 0.3z&=0.11\\ 0.2x - 0.8y - 0.9z&=-0.32\end{align}[/latex]

43. [latex]\begin{align}0.5x+0.2y - 0.3z&=1\\ 0.4x - 0.6y+0.7z&=0.8\\ 0.3x - 0.1y - 0.9z&=0.6\end{align}[/latex]

44. [latex]\begin{align}0.3x+0.3y+0.5z&=0.6\\ 0.4x+0.4y+0.4z&=1.8\\ 0.4x+0.2y+0.1z&=1.6\end{align}[/latex]

45. [latex]\begin{align}0.8x+0.8y+0.8z&=2.4\\ 0.3x - 0.5y+0.2z&=0\\ 0.1x+0.2y+0.3z&=0.6\end{align}[/latex]

For the following exercises, solve the system for [latex]x,y[/latex], and [latex]z[/latex].

46. [latex]\begin{align}x+y+z&=3 \\ \frac{x - 1}{2}+\frac{y - 3}{2}+\frac{z+1}{2}&=0 \\ \frac{x - 2}{3}+\frac{y+4}{3}+\frac{z - 3}{3}&=\frac{2}{3} \end{align}[/latex]

47. [latex]\begin{align}5x - 3y-\frac{z+1}{2}&=\frac{1}{2} \\ 6x+\frac{y - 9}{2}+2z&=-3 \\ \frac{x+8}{2}-4y+z&=4 \end{align}[/latex]

48. [latex]\begin{align}\frac{x+4}{7}-\frac{y - 1}{6}+\frac{z+2}{3}&=1\\ \frac{x - 2}{4}+\frac{y+1}{8}-\frac{z+8}{12}&=0\\ \frac{x+6}{3}-\frac{y+2}{3}+\frac{z+4}{2}7&=3\end{align}[/latex]

49. [latex]\begin{align}\frac{x - 3}{6}+\frac{y+2}{2}-\frac{z - 3}{3}&=2\\ \frac{x+2}{4}+\frac{y - 5}{2}+\frac{z+4}{2}&=1\\ \frac{x+6}{2}-\frac{y - 3}{2}+z+1&=9\end{align}[/latex]

50. [latex]\begin{align}\frac{x - 1}{3}+\frac{y+3}{4}+\frac{z+2}{6}&=1 \\ 4x+3y - 2z&=11 \\ 0.02x+0.015y - 0.01z&=0.065 \end{align}[/latex]