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

2. Can a system of two linear equations have more than two solutions? Explain why or why not.

3. Given a system of equations, explain at least two different methods of solving that system.

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

4. [latex]\begin{array}{c}5x-y=4\\ x+6y=2\end{array}[/latex] and [latex]\left(4,0\right)[/latex]

5. [latex]\begin{array}{l}-3x - 5y=13\hfill \\ -x+4y=10\hfill \end{array}[/latex] and [latex]\left(-6,1\right)[/latex]

6. [latex]\begin{array}{c}3x+7y=1\\ 2x+4y=0\end{array}[/latex] and [latex]\left(2,3\right)[/latex]

7. [latex]\begin{array}{l}-2x+5y=7\hfill \\ \text{ }2x+9y=7\hfill \end{array}[/latex] and [latex]\left(-1,1\right)[/latex]

8. [latex]\begin{array}{c}x+8y=43\\ 3x - 2y=-1\end{array}[/latex] and [latex]\left(3,5\right)[/latex]

For the following exercises, solve each system by substitution.

9. [latex]\begin{array}{l}\text{ }x+3y=5\hfill \\ 2x+3y=4\hfill \end{array}[/latex]

10. [latex]\begin{array}{l}\text{ }3x - 2y=18\hfill \\ 5x+10y=-10\hfill \end{array}[/latex]

11. [latex]\begin{array}{l}4x+2y=-10\\ 3x+9y=0\end{array}[/latex]

12. [latex]\begin{array}{l}2x+4y=-3.8\\ 9x - 5y=1.3\end{array}[/latex]

13. [latex]\begin{array}{l}\hfill \\ \begin{array}{l}\\ \begin{array}{l}-2x+3y=1.2\hfill \\ -3x - 6y=1.8\hfill \end{array}\end{array}\hfill \end{array}[/latex]

14. [latex]\begin{array}{l}\text{ }x - 0.2y=1\hfill \\ -10x+2y=5\hfill \end{array}[/latex]

15. [latex]\begin{array}{l}\text{ }3x+5y=9\hfill \\ 30x+50y=-90\hfill \end{array}[/latex]

16. [latex]\begin{array}{l}\hfill \\ \begin{array}{l}\text{ }-3x+y=2\hfill \\ 12x - 4y=-8\hfill \end{array}\hfill \end{array}[/latex]

17. [latex]\begin{array}{l}\frac{1}{2}x+\frac{1}{3}y=16\\ \frac{1}{6}x+\frac{1}{4}y=9\end{array}[/latex]

18. [latex]\begin{array}{l}\\ \begin{array}{l}-\frac{1}{4}x+\frac{3}{2}y=11\hfill \\ -\frac{1}{8}x+\frac{1}{3}y=3\hfill \end{array}\end{array}[/latex]

For the following exercises, solve each system by addition.

19. [latex]\begin{array}{l}\hfill \\ \begin{array}{l}-2x+5y=-42\hfill \\ \text{ }7x+2y=30\hfill \end{array}\hfill \end{array}[/latex]

20. [latex]\begin{array}{l}6x - 5y=-34\\ 2x+6y=4\end{array}[/latex]

21. [latex]\begin{array}{l}\text{ }5x-y=-2.6\hfill \\ -4x - 6y=1.4\hfill \end{array}[/latex]

22. [latex]\begin{array}{l}7x - 2y=3\\ 4x+5y=3.25\end{array}[/latex]

23. [latex]\begin{array}{l}\hfill \\ \begin{array}{l}\text{ }\mathrm{-x}+2y=-1\hfill \\ 5x - 10y=6\hfill \end{array}\hfill \end{array}[/latex]

24. [latex]\begin{array}{l}\text{ }7x+6y=2\hfill \\ -28x - 24y=-8\hfill \end{array}[/latex]

25. [latex]\begin{array}{l}\frac{5}{6}x+\frac{1}{4}y=0\\ \frac{1}{8}x-\frac{1}{2}y=-\frac{43}{120}\end{array}[/latex]

26. [latex]\begin{array}{l}\text{ }\frac{1}{3}x+\frac{1}{9}y=\frac{2}{9}\hfill \\ -\frac{1}{2}x+\frac{4}{5}y=-\frac{1}{3}\hfill \end{array}[/latex]

27. [latex]\begin{array}{l}\hfill \\ \begin{array}{l}-0.2x+0.4y=0.6\hfill \\ \text{ }x - 2y=-3\hfill \end{array}\hfill \end{array}[/latex]

28. [latex]\begin{array}{l}\begin{array}{l}\\ -0.1x+0.2y=0.6\end{array}\hfill \\ \text{ }5x - 10y=1\hfill \end{array}[/latex]

For the following exercises, solve each system by any method.

29. [latex]\begin{array}{l}5x+9y=16\hfill \\ \text{ }x+2y=4\hfill \end{array}[/latex]

30. [latex]\begin{array}{l}6x - 8y=-0.6\\ 3x+2y=0.9\end{array}[/latex]

31. [latex]\begin{array}{l}5x - 2y=2.25\\ 7x - 4y=3\end{array}[/latex]

32. [latex]\begin{array}{l}\begin{array}{l}\hfill \\ \text{ }x-\frac{5}{12}y=-\frac{55}{12}\hfill \end{array}\hfill \\ -6x+\frac{5}{2}y=\frac{55}{2}\hfill \end{array}[/latex]

33. [latex]\begin{array}{l}\\ \begin{array}{l}7x - 4y=\frac{7}{6}\hfill \\ 2x+4y=\frac{1}{3}\hfill \end{array}\end{array}[/latex]

34. [latex]\begin{array}{l}3x+6y=11\\ 2x+4y=9\end{array}[/latex]

35. [latex]\begin{array}{l}\text{ }\frac{7}{3}x-\frac{1}{6}y=2\hfill \\ -\frac{21}{6}x+\frac{3}{12}y=-3\hfill \end{array}[/latex]

36. [latex]\begin{array}{l}\frac{1}{2}x+\frac{1}{3}y=\frac{1}{3}\\ \frac{3}{2}x+\frac{1}{4}y=-\frac{1}{8}\end{array}[/latex]

37. [latex]\begin{array}{l}2.2x+1.3y=-0.1\\ 4.2x+4.2y=2.1\end{array}[/latex]

38. [latex]\begin{array}{l}\text{ }0.1x+0.2y=2\hfill \\ 0.35x - 0.3y=0\hfill \end{array}[/latex]

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

39. [latex]\begin{array}{l}3x-y=0.6\\ x - 2y=1.3\end{array}[/latex]

40. [latex]\begin{array}{l}\begin{array}{l}\\ -x+2y=4\end{array}\hfill \\ \text{ }2x - 4y=1\hfill \end{array}[/latex]

41. [latex]\begin{array}{l}\text{ }x+2y=7\hfill \\ 2x+6y=12\hfill \end{array}[/latex]

42. [latex]\begin{array}{l}3x - 5y=7\hfill \\ \text{ }x - 2y=3\hfill \end{array}[/latex]

43. [latex]\begin{array}{l}\text{ }3x - 2y=5\hfill \\ -9x+6y=-15\hfill \end{array}[/latex]

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

44. [latex]\begin{array}{l}\text{ }0.1x+0.2y=0.3\hfill \\ -0.3x+0.5y=1\hfill \end{array}[/latex]

45. [latex]\begin{array}{l}\hfill \\ \begin{array}{l}-0.01x+0.12y=0.62\hfill \\ 0.15x+0.20y=0.52\hfill \end{array}\hfill \end{array}[/latex]

46. [latex]\begin{array}{l}0.5x+0.3y=4\hfill \\ 0.25x - 0.9y=0.46\hfill \end{array}[/latex]

47. [latex]\begin{array}{l}0.15x+0.27y=0.39\hfill \\ -0.34x+0.56y=1.8\hfill \end{array}[/latex]

48. [latex]\begin{array}{l}\begin{array}{l}\\ -0.71x+0.92y=0.13\end{array}\hfill \\ 0.83x+0.05y=2.1\hfill \end{array}[/latex]

For the following exercises, solve each system in terms of [latex]A,B,C,D,E,\text{}[/latex] and [latex]F[/latex] where [latex]A-F[/latex] are nonzero numbers. Note that [latex]A\ne B[/latex] and [latex]AE\ne BD[/latex].

49. [latex]\begin{array}{l}x+y=A\\ x-y=B\end{array}[/latex]

50. [latex]\begin{array}{l}x+Ay=1\\ x+By=1\end{array}[/latex]

51. [latex]\begin{array}{l}Ax+y=0\\ Bx+y=1\end{array}[/latex]

52. [latex]\begin{array}{l}Ax+By=C\\ x+y=1\end{array}[/latex]

53. [latex]\begin{array}{l}Ax+By=C\\ Dx+Ey=F\end{array}[/latex]