Exercises: Solve Systems of Equations by Graphing (4.1)

Domenic Spilotro, MSc

Exercises: Solve Systems of Equations by Graphing (4.1)

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

Instructions: For questions 1-8, determine if the following points are solutions to the given system of equations.

1. ${\begin{matrix} 2 x - 6 y = 0 \\ 3 x - 4 y = 5 \end{matrix}$

a. $(3, 1)$
b. $(- 3, 4)$

Solution

a. yes
b. no

2. ${\begin{matrix} 7 x - 4 y = - 1 \\ - 3 x - 2 y = 1 \end{matrix}$

a. $(1, 2)$
b. $(1, - 2)$

3. ${\begin{matrix} 2 x + y = 5 \\ x + y = 1 \end{matrix}$

a. $(4, 3)$
b. $(2, 0)$

Solution

a. yes
b. no

4. ${\begin{matrix} - 3 x + y = 8 \\ - x + 2 y = - 9 \end{matrix}$

a. $(- 5, - 7)$
b. $(- 5, 7)$

5. ${\begin{matrix} x + y = 2 \\ y = \frac{3}{4} x \end{matrix}$

a. $(\frac{8}{7}, \frac{6}{7})$
b. $(1, \frac{3}{4})$

Solution

a. yes
b. no

6. ${\begin{matrix} x + y = 1 \\ y = \frac{2}{5} x \end{matrix}$

a. $(\frac{5}{7}, \frac{2}{7})$
b. $(5, 2)$

7. ${\begin{matrix} x + 5 y = 10 \\ y = \frac{3}{5} x + 1 \end{matrix}$

a. $(- 10, 4)$
b. $(\frac{5}{4}, \frac{7}{4})$

Solution

a. no
b. yes

8. ${\begin{matrix} x + 3 y = 9 \\ y = \frac{2}{3} x - 2 \end{matrix}$

a. $(- 6, 5)$
b. $(5, \frac{4}{3})$

Exercises: Solve a System of Linear Equations by Graphing

Instructions: For questions 9-50, solve the following systems of equations by graphing.

9. ${\begin{matrix} 3 x + y = - 3 \\ 2 x + 3 y = 5 \end{matrix}$

Solution

$(- 2, 3)$

10. ${\begin{matrix} - x + y = 2 \\ 2 x + y = - 4 \end{matrix}$

11. ${\begin{matrix} - 3 x + y = - 1 \\ 2 x + y = 4 \end{matrix}$

Solution

$(1, 2)$

12. ${\begin{matrix} - 2 x + 3 y = - 3 \\ x + y = 4 \end{matrix}$

13. ${\begin{matrix} y = x + 2 \\ y = - 2 x + 2 \end{matrix}$

Solution

$(0, 2)$

14. ${\begin{matrix} y = x - 2 \\ y = - 3 x + 2 \end{matrix}$

15. ${\begin{matrix} y = \frac{3}{2} x + 1 \\ y = - \frac{1}{2} x + 5 \end{matrix}$

Solution

$(2, 4)$

16. ${\begin{matrix} y = \frac{2}{3} x - 2 \\ y = - \frac{1}{3} x - 5 \end{matrix}$

17. ${\begin{matrix} - x + y = - 3 \\ 4 x + 4 y = 4 \end{matrix}$

Solution

$(2, - 1)$

18. ${\begin{matrix} x - y = 3 \\ 2 x - y = 4 \end{matrix}$

19. ${\begin{matrix} - 3 x + y = - 1 \\ 2 x + y = 4 \end{matrix}$

Solution

$(1, 2)$

20. ${\begin{matrix} - 3 x + y = - 2 \\ 4 x - 2 y = 6 \end{matrix}$

21. ${\begin{matrix} x + y = 5 \\ 2 x - y = 4 \end{matrix}$

Solution

$(3, 2)$

22. ${\begin{matrix} x - y = 2 \\ 2 x - y = 6 \end{matrix}$

23. ${\begin{matrix} x + y = 2 \\ x - y = 0 \end{matrix}$

Solution

$(1, 1)$

24. ${\begin{matrix} x + y = 6 \\ x - y = - 8 \end{matrix}$

25. ${\begin{matrix} x + y = - 5 \\ x - y = 3 \end{matrix}$

Solution

$(- 1, - 4)$

26. ${\begin{matrix} x + y = 4 \\ x - y = 0 \end{matrix}$

27. ${\begin{matrix} x + y = - 4 \\ - x + 2 y = - 2 \end{matrix}$

Solution

$(3, 3)$

28. ${\begin{matrix} - x + 3 y = 3 \\ x + 3 y = 3 \end{matrix}$

29. ${\begin{matrix} - 2 x + 3 y = 3 \\ x + 3 y = 12 \end{matrix}$

Solution

$(- 5, 6)$

30. ${\begin{matrix} 2 x - y = 4 \\ 2 x + 3 y = 12 \end{matrix}$

31. ${\begin{matrix} 2 x + 3 y = 6 \\ y = - 2 \end{matrix}$

Solution

$(6, - 2)$

32. ${\begin{matrix} - 2 x + y = 2 \\ y = 4 \end{matrix}$

33. ${\begin{matrix} x - 3 y = - 3 \\ y = 2 \end{matrix}$

Solution

$(3, 2)$

34. ${\begin{matrix} 2 x - 2 y = 8 \\ y = - 3 \end{matrix}$

35. ${\begin{matrix} 2 x - y = - 1 \\ x = 1 \end{matrix}$

Solution

$(1, 3)$

36. ${\begin{matrix} x + 2 y = 2 \\ x = - 2 \end{matrix}$

37. ${\begin{matrix} x - 3 y = - 6 \\ x = - 3 \end{matrix}$

Solution

$(- 3, 1)$

38. ${\begin{matrix} x + y = 4 \\ x = 1 \end{matrix}$

39. ${\begin{matrix} 4 x - 3 y = 8 \\ 8 x - 6 y = 14 \end{matrix}$

Solution

no solution

40. ${\begin{matrix} x + 3 y = 4 \\ - 2 x - 6 y = 3 \end{matrix}$

41. ${\begin{matrix} - 2 x + 4 y = 4 \\ y = \frac{1}{2} x \end{matrix}$

Solution

no solution

42. ${\begin{matrix} 3 x + 5 y = 10 \\ y = - \frac{3}{5} x + 1 \end{matrix}$

43. ${\begin{matrix} x = - 3 y + 4 \\ 2 x + 6 y = 8 \end{matrix}$

Solution

no solution

44. ${\begin{matrix} 4 x = 3 y + 7 \\ 8 x - 6 y = 14 \end{matrix}$

45. ${\begin{matrix} 2 x + y = 6 \\ - 8 x - 4 y = - 24 \end{matrix}$

Solution

infinitely many solutions

46. ${\begin{matrix} 5 x + 2 y = 7 \\ - 10 x - 4 y = - 14 \end{matrix}$

47. ${\begin{matrix} x + 3 y = - 6 \\ 4 y = - \frac{4}{3} x - 8 \end{matrix}$

Solution

infinitely many solutions

48. ${\begin{matrix} - x + 2 y = - 6 \\ y = - \frac{1}{2} x - 1 \end{matrix}$

49. ${\begin{matrix} - 3 x + 2 y = - 2 \\ y = - x + 4 \end{matrix}$

Solution

$(2, 2)$

50. ${\begin{matrix} - x + 2 y = - 2 \\ y = - x - 1 \end{matrix}$

Exercises: Determine the Number of Solutions of a Linear System

Instructions: For questions 51-62, determine the number of solutions and then classify the system of equations without graphing.

51. ${\begin{matrix} y = \frac{2}{3} x + 1 \\ - 2 x + 3 y = 5 \end{matrix}$

Solution

$0$ solutions

52. ${\begin{matrix} y = \frac{1}{3} x + 2 \\ x - 3 y = 9 \end{matrix}$

53. ${\begin{matrix} y = - 2 x + 1 \\ 4 x + 2 y = 8 \end{matrix}$

Solution

$0$ solutions

54. ${\begin{matrix} y = 3 x + 4 \\ 9 x - 3 y = 18 \end{matrix}$

55. ${\begin{matrix} y = \frac{2}{3} x + 1 \\ 2 x - 3 y = 7 \end{matrix}$

Solution

no solutions, inconsistent, independent

56. ${\begin{matrix} 3 x + 4 y = 12 \\ y = - 3 x - 1 \end{matrix}$

57. ${\begin{matrix} 4 x + 2 y = 10 \\ 4 x - 2 y = - 6 \end{matrix}$

Solution

consistent, $1$ solution

58. ${\begin{matrix} 5 x + 3 y = 4 \\ 2 x - 3 y = 5 \end{matrix}$

59. ${\begin{matrix} y = - \frac{1}{2} x + 5 \\ x + 2 y = 10 \end{matrix}$

Solution

infinitely many solutions

60. ${\begin{matrix} y = x + 1 \\ - x + y = 1 \end{matrix}$

61. ${\begin{matrix} y = 2 x + 3 \\ 2 x - y = - 3 \end{matrix}$

Solution

infinitely many solutions

62. ${\begin{matrix} 5 x - 2 y = 10 \\ y = \frac{5}{2} x - 5 \end{matrix}$

Exercises: Solve Applications of Systems of Equations by Graphing

Instructions: For questions 63-66, solve.

63. Molly is making strawberry infused water. For each ounce of strawberry juice, she uses three times as many ounces of water. How many ounces of strawberry juice and how many ounces of water does she need to make $64$ ounces of strawberry infused water?

Solution

Molly needs $16$ ounces of strawberry juice and $48$ ounces of water.

64. Jamal is making a snack mix that contains only pretzels and nuts. For every ounce of nuts, he will use $2$ ounces of pretzels. How many ounces of pretzels and how many ounces of nuts does he need to make $45$ ounces of snack mix?

65. Enrique is making a party mix that contains raisins and nuts. For each ounce of nuts, he uses twice the amount of raisins. How many ounces of nuts and how many ounces of raisins does he need to make $24$ ounces of party mix?

Solution

Enrique needs $8$ ounces of nuts and $16$ ounces of water.

66. Owen is making lemonade from concentrate. The number of quarts of water he needs is $4$ times the number of quarts of concentrate. How many quarts of water and how many quarts of concentrate does Owen need to make $100$ quarts of lemonade?

Exercises: Everyday Math

Instructions: For questions 67-68, answer the given everyday math word problems.

67. Leo is planning his spring flower garden. He wants to plant tulip and daffodil bulbs. He will plant 6 times as many daffodil bulbs as tulip bulbs. If he wants to plant $350$ bulbs, how many tulip bulbs and how many daffodil bulbs should he plant?

Solution

Leo should plant $50$ tulips and $300$ daffodils.

68. A marketing company surveys $1, 200$ people. They surveyed twice as many females as males. How many males and females did they survey?

Exercises: Writing Exercises

Instructions: For questions 69-70, answer the given writing exercises.

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

Solution

Given that it is only known that the slopes of both linear equations are the same, there are either no solutions (the graphs of the equations are parallel) or infinitely many.

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

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Fanshawe Pre-Health Sciences Mathematics 1 Copyright © 2022 by Domenic Spilotro, MSc is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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