
# 4.1E: Exercises


## Practice Makes Perfect

Determine Whether an Ordered Pair is a Solution of a System of Equations

In the following exercises, determine if the following points are solutions to the given system of equations.

1. $$\left\{ \begin{array} {l} 2x−6y=0 \\ 3x−4y=5 \end{array} \right.$$

ⓐ $$(3,1)$$
ⓑ $$(−3,4)$$

ⓐ yes ⓑ no

2. $$\left\{ \begin{array} {l} −3x+y=8 \\ −x+2y=−9 \end{array} \right.$$

ⓐ $$(−5,−7)$$
ⓑ $$(−5,7)$$

3. $$\left\{ \begin{array} {l} x+y=2 \\ y=\frac{3}{4}x \end{array} \right.$$

ⓐ $$(87,67)$$
ⓑ $$(1,34)$$

ⓐ yes ⓑ no

4. $$\left\{ \begin{array} {l} 2x+3y=6 \\ y=\frac{2}{3}x+2 \end{array} \right.$$

ⓐ $$(−6,2)$$
ⓑ $$(−3,4)$$

Solve a System of Linear Equations by Graphing

In the following exercises, solve the following systems of equations by graphing.

5. $$\left\{ \begin{array} {l} 3x+y=−3 \\ 2x+3y=5 \end{array} \right.$$

$$(−3,2)$$

6. $$\left\{ \begin{array} {l} −x+y=2 \\ 2x+y=−4 \end{array} \right.$$

7. $$\left\{ \begin{array} {l} y=x+2 \\ y=−2x+2 \end{array} \right.$$

$$(0,2)$$

8. $$\left\{ \begin{array} {l} y=x−2 \\ y=−3x+2 \end{array} \right.$$

9. $$\left\{ \begin{array} {l} y=\frac{3}{2}x+1 \\ y=−\frac{1}{2}x+5 \end{array} \right.$$

$$(2,4)$$

10. $$\left\{ \begin{array} {l} y=\frac{2}{3}x−2 \\ y=−\frac{1}{3}x−5 \end{array} \right.$$

11. $$\left\{ \begin{array} {l} x+y=−4 \\ −x+2y=−2 \end{array} \right.$$

$$(−2,2)$$

12. $$\left\{ \begin{array} {l} −x+3y=3 \\ x+3y=3 \end{array} \right.$$

13. $$\left\{ \begin{array} {l} −2x+3y=3 \\ x+3y=12 \end{array} \right.$$

$$(3,3)$$

14. $$\left\{ \begin{array} {l} 2x−y=4 \\ 2x+3y=12 \end{array} \right.$$

15. $$\left\{ \begin{array} {l} x+3y=−6 \\ y=−\frac{4}{3}x+4 \end{array} \right.$$

$$(6,−4)$$

16. $$\left\{ \begin{array} {l} −x+2y=−6 \\ y=−\frac{1}{2}x−1 \end{array} \right.$$

17. $$\left\{ \begin{array} {l} −2x+4y=4 \\ y=12x \end{array} \right.$$

no solution

18. $$\left\{ \begin{array} {l} 3x+5y=10 \\ y=−\frac{3}{5}x+1 \end{array} \right.$$

19. $$\left\{ \begin{array} {l} 4x−3y=8 \\ 8x−6y=14 \end{array} \right.$$

no solution

20. $$\left\{ \begin{array} {l} x+3y=4 \\ −2x−6y=3 \end{array} \right.$$

21. $$\left\{ \begin{array} {l} x=−3y+4 \\ 2x+6y=8 \end{array} \right.$$

infinite solutions with solution set: $$\big\{ (x,y) | 2x+6y=8 \big\}$$

22. $$\left\{ \begin{array} {l} 4x=3y+7 \\ 8x−6y=14 \end{array} \right.$$

23. $$\left\{ \begin{array} {l} 2x+y=6 \\ −8x−4y=−24 \end{array} \right.$$

infinite solutions with solution set: $$\big\{ (x,y) | 2x+y=6 \big\}$$

24. $$\left\{ \begin{array} {l} 5x+2y=7 \\ −10x−4y=−14 \end{array} \right.$$

Without graphing, determine the number of solutions and then classify the system of equations.

25. $$\left\{ \begin{array} {l} y=\frac{2}{3}x+1 \\ −2x+3y=5 \end{array} \right.$$

1 point, consistent and independent

26. $$\left\{ \begin{array} {l} y=\frac{3}{2}x+1 \\ 2x−3y=7 \end{array} \right.$$

27. $$\left\{ \begin{array} {l} 5x+3y=4 \\ 2x−3y=5 \end{array} \right.$$

1 point, consistent and independent

28. $$\left\{ \begin{array} {l} y=−12x+5 \\ x+2y=10 \end{array} \right.$$

29. $$\left\{ \begin{array} {l} 5x−2y=10 \\ y=52x−5 \end{array} \right.$$

infinite solutions, consistent, dependent

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

30. $$\left\{ \begin{array} {l} 2x+y=−4 \\ 3x−2y=−6\end{array} \right.$$

31. $$\left\{ \begin{array} {l} 2x+y=−2\\ 3x−y=7 \end{array} \right.$$

$$(1,−4)$$

32. $$\left\{ \begin{array} {l} x−2y=−5 \\ 2x−3y=−4 \end{array} \right.$$

33. $$\left\{ \begin{array} {l} x−3y=−9 \\ 2x+5y=4 \end{array} \right.$$

$$(−3,2)$$

34. $$\left\{ \begin{array} {l} 5x−2y=−6 \\ y=3x+3 \end{array} \right.$$

35. $$\left\{ \begin{array} {l} −2x+2y=6 \\ y=−3x+1 \end{array} \right.$$

$$(−1/2,5/2)$$

36. $$\left\{ \begin{array} {l} 2x+5y=1 \\ y=\frac{1}{3}x−2 \end{array} \right.$$

37. $$\left\{ \begin{array} {l} 3x+4y=1 \\ y=−\frac{2}{5}x+2 \end{array} \right.$$

$$(−5,4)$$

38. $$\left\{ \begin{array} {l} 2x+y=5 \\ x−2y=−15 \end{array} \right.$$

39. $$\left\{ \begin{array} {l} 4x+y=10 \\ x−2y=−20 \end{array} \right.$$

$$(0,10)$$

40. $$\left\{ \begin{array} {l} y=−2x−1 \\ y=−\frac{1}{3}x+4 \end{array} \right.$$

41. $$\left\{ \begin{array} {l} y=x−6 \\ y=−\frac{3}{2}x+4 \end{array} \right.$$

$$(4,−2)$$

42. $$\left\{ \begin{array} {l} x=2y \\ 4x−8y=0 \end{array} \right.$$

43. $$\left\{ \begin{array} {l} 2x−16y=8 \\ −x−8y=−4 \end{array} \right.$$

$$(4,0)$$

44. $$\left\{ \begin{array} {l} y=\frac{7}{8}x+4 \\ −7x+8y=6 \end{array} \right.$$

45. $$\left\{ \begin{array} {l} y=−\frac{2}{3}x+5 \\ 2x+3y=11 \end{array} \right.$$

no solution

Solve a System of Equations by Elimination

In the following exercises, solve the systems of equations by elimination.

46. $$\left\{ \begin{array} {l} 5x+2y=2 \\ −3x−y=0 \end{array} \right.$$

47. $$\left\{ \begin{array} {l} 6x−5y=−1 \\ 2x+y=13 \end{array} \right.$$

$$(4,5)$$

48. $$\left\{ \begin{array} {l} 2x−5y=7 \\ 3x−y=17 \end{array} \right.$$

49. $$\left\{ \begin{array} {l} 5x−3y=−1 \\ 2x−y=2 \end{array} \right.$$

$$(7,12)$$

50. $$\left\{ \begin{array} {l} 3x−5y=−9 \\ 5x+2y=16 \end{array} \right.$$

51. $$\left\{ \begin{array} {l} 4x−3y=3 \\ 2x+5y=−31 \end{array} \right.$$

$$(−3,−5)$$

52. $$\left\{ \begin{array} {l} 3x+8y=−3 \\ 2x+5y=−3 \end{array} \right.$$

53. $$\left\{ \begin{array} {l} 11x+9y=−5 \\ 7x+5y=−1 \end{array} \right.$$

$$(2,−3)$$

54. $$\left\{ \begin{array} {l} 3x+8y=67 \\ 5x+3y=60 \end{array} \right.$$

55. $$\left\{ \begin{array} {l} 2x+9y=−4 \\ 3x+13y=−7 \end{array} \right.$$

$$(−11,2)$$

56. $$\left\{ \begin{array} {l} \frac{1}{3}x−y=−3 \\ x+\frac{5}{2}y=2 \end{array} \right.$$

57. $$\left\{ \begin{array} {l} x+\frac{1}{2}y=\frac{3}{2} \\ \frac{1}{5}x−\frac{1}{5}y=3 \end{array} \right.$$

$$(6/−9,24/7)$$

58. $$\left\{ \begin{array} {l} x+\frac{1}{3}y=−1 \\ \frac{1}{3}x+\frac{1}{2}y=1 \end{array} \right.$$

59. $$\left\{ \begin{array} {l} \frac{1}{3}x−y=−3 \\ \frac{2}{3}x+\frac{5}{2}y=3 \end{array} \right.$$

$$(−3,2)$$

60. $$\left\{ \begin{array} {l} 2x+y=3 \\ 6x+3y=9 \end{array} \right.$$

61. $$\left\{ \begin{array} {l} x−4y=−1 \\ −3x+12y=3 \end{array} \right.$$

infinitely many solutions with solution set: $$\big\{ (x,y) | x−4y=−1 \big\}$$

62. $$\left\{ \begin{array} {l} −3x−y=8 \\ 6x+2y=−16 \end{array} \right.$$

63. $$\left\{ \begin{array} {l} 4x+3y=2 \\ 20x+15y=10 \end{array} \right.$$

infinitely many solutions with solution set: $$\big\{ (x,y) | 4x+3y=2 \big\}$$

Choose the Most Convenient Method to Solve a System of Linear Equations

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

64.
ⓐ $$\left\{ \begin{array} {l} 8x−15y=−32 \\ 6x+3y=−5 \end{array} \right.$$

ⓑ $$\left\{ \begin{array} {l} x=4y−3 \\ 4x−2y=−6 \end{array} \right.$$

65.
ⓐ $$\left\{ \begin{array} {l} y=7x−5 \\ 3x−2y=16 \end{array} \right.$$

ⓑ $$\left\{ \begin{array} {l} 12x−5y=−42 \\ 3x+7y=−15 \end{array} \right.$$

ⓐ substitution ⓑ elimination

66.
ⓐ $$\left\{ \begin{array} {l} y=4x+95 \\ x−2y=−21 \end{array} \right.$$

ⓑ $$\left\{ \begin{array} {l} 9x−4y=24 \\ 3x+5y=−14 \end{array} \right.$$

67.
ⓐ $$\left\{ \begin{array} {l} 14x−15y=−30 \\ 7x+2y=10 \end{array} \right.$$

ⓑ $$\left\{ \begin{array} {l} x=9y−11 \\ 2x−7y=−27 \end{array} \right.$$

ⓐ elimination ⓑ substitution

## Writing Exercises

68. In a system of linear equations, the two equations have the same intercepts. Describe the possible solutions to the system.

69. Solve the system of equations by substitution and explain all your steps in words: $$\left\{ \begin{array} {l} 3x+y=1 \\ 2x=y−8 \end{array} \right.$$

70. Solve the system of equations by elimination and explain all your steps in words: $$\left\{ \begin{array} {l} 5x+4y=10 \\ 2x=3y+27 \end{array} \right.$$

71. Solve the system of equations $$\left\{ \begin{array} {l} x+y=10 \\ x−y=6 \end{array} \right.$$

ⓐ by graphing ⓑ by substitution
ⓒ Which method do you prefer? Why?