1. Linear System of Equations
Classify Systems
Independent
Dependent
Inconsistent
Methods for Solving
Tables
Graphing
Substitution
Elimination
Matrices
𝑦=𝑥+3 𝑥+𝑦=1
(−1, 2)
𝒙+(𝒙+𝟑)=𝟏
𝟐𝒙=−𝟐
𝒙=−𝟏
𝒚= −𝟏 +𝟑=𝟐
−𝒙+𝒚=𝟑
𝒙+𝒚=𝟏
𝟐𝒚=𝟒
𝒚=𝟐
𝟐=𝒙+𝟑
−𝟏=𝒙 𝑥
𝑦=𝑥+3
𝑥+𝑦=1 −2 1 3 −1 2
−𝟏 𝟏 𝟏 𝟏 | 𝟑 𝟏
𝟏 𝟎 𝟎 𝟏 −𝟏 𝟐

2. Linear System of Inequalities
Maximize: 3𝑥+2𝑦
−𝒙+𝟐𝒚≤𝟒 𝒙+𝒚<𝟖 𝒙≥𝟐 𝒚≥𝟏
(2, 1) =8
(2, 2)
=10
(4, 4)
=20
(7, 1)
=23
Pg. 187 #1 – 13

3. Linear Systems 10 20 30 40 50 Classifying Systems
Solving 2 Variable Systems
Solving 3 Variable Systems
Inequalities
Modeling (Application) 10 20 30 40 50

4. Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent)
5𝑥+3𝑦=9 𝑦=− 3 5 𝑥+3
Answer

5. Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent)
5𝑥+3𝑦=9
3𝑦=−5𝑥+9
𝑦=− 5 3 𝑥+3
5𝑥+3𝑦=9 𝑦=− 3 5 𝑥+3
Independent

6. Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent)
12𝑥−4𝑦=20 𝑦=3𝑥+3
Answer

7. Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent)
12𝑥−4𝑦=20
−4𝑦=−12𝑥+20
𝑦=3𝑥−5
12𝑥−4𝑦=20 𝑦=3𝑥+3
Inconsistent

8. Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent)
2𝑥+6𝑦=12 𝑦=− 1 3 𝑥+2
Answer

9. Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent)
2𝑥+6𝑦=12 𝑦=− 1 3 𝑥+2
2𝑥+6𝑦=12
6𝑦=−2𝑥+12
𝑦=− 1 3 𝑥+2
Dependent

10. Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent)
−12𝑥+2𝑦=−15 18𝑥−3𝑦=27
Answer

11. Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent)
−12𝑥+2𝑦=−15 18𝑥−3𝑦=27
−12𝑥+2𝑦=−15
2𝑦=12𝑥−15
𝑦=6𝑥− 15 2
18𝑥−3𝑦=27
−3𝑦=−18𝑥+27
𝑦=6𝑥−9
Inconsistent

12. Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent)
15𝑥+6𝑦=6 10𝑥+4𝑦=4
Answer

13. Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent)
15𝑥+6𝑦=6 10𝑥+4𝑦=4
15𝑥+6𝑦=6
6𝑦=−15𝑥+6
𝑦=− 5 2 𝑥+1
10𝑥+4𝑦=4
4𝑦=−10𝑥+4
𝑦=− 5 2 𝑥+1
Dependent

14. Solve the system by: Graphing
𝑦=2𝑥−4 𝑥−4𝑦=−3
Answer

15. Solve the system by: Graphing
𝑦=2𝑥−4 𝑥−4𝑦=−12
(4, 4)

16. Solve the system by: Substitution
−2𝑥−𝑦=−9 𝑦=−5𝑥+15
Answer

17. Solve the system by: Substitution
−2𝑥−𝑦=−9 𝑦=−5𝑥+15
−2𝑥−𝑦=−9
−2𝑥−(−5𝑥+15)=−9
−2𝑥+5𝑥−15=−9
3𝑥=6
𝑥=2
𝑦=−5𝑥+15
𝑦=−
𝑦=5
(2, 5)

18. Solve the system by: Elimination
−8𝑥−7𝑦=−28 5𝑥+6𝑦=24
Answer

19. Solve the system by: Elimination
−8𝑥−7𝑦=−28 5𝑥+6𝑦=24
𝟔 −8𝑥−7𝑦=−28
𝟕(5𝑥+6𝑦=24)
−48𝑥−42𝑦=−168
35𝑥+42𝑦=168
−13𝑥=0
𝑥=0
5𝑥+6𝑦=24
5 0 +6𝑦=24
6𝑦=24
𝑦=4
(0, 4)

20. Solve the system by: Your Choice
6𝑦+11+𝑥=0 8𝑥=−4−6𝑦
Answer

21. Solve the system by: Your Choice
6𝑦+11+𝑥=0 8𝑥=−4−6𝑦
𝑥+6𝑦=−11
−8𝑥−6𝑦=4
−7𝑥=−7
𝑥=1
6𝑦+11+𝑥=0
6𝑦 =0
6𝑦=−12
𝑦=−2
(1, −2)

22. Solve the system by: Your Choice
−4𝑦=8𝑥+12 0=18𝑦+12𝑥−90
Answer

23. Solve the system by: Your Choice
−4𝑦=8𝑥+12 0=18𝑦+12𝑥−90
0=18𝑦+12𝑥−90
0=18 −2𝑥−3 +12𝑥−90
0=−36𝑥−54+12𝑥−90
144=−24𝑥
−6=𝑥
−4𝑦=8𝑥+12
𝑦=−2𝑥−3
𝑦=−2 −6 −3
𝑦=9
(−6, 9)

24. Write a matrix to represent the system.
6𝑥+2𝑦+𝑧=30 −3𝑥+3𝑧=0 −2𝑥+5𝑦+4𝑧=3
Answer

25. Write a matrix to represent the system.
6𝑥+2𝑦+𝑧=30 −3𝑥+3𝑧=0 −2𝑥+5𝑦+4𝑧=3
6 2 1 −3 0 3 −

26. What system is represented by the matrix:−
Answer

27. What system is represented by the matrix:−
2𝑥+7𝑦+𝑧=9 3𝑥−2𝑦=6 𝑥+2𝑦+𝑧=0

28. Solve the system by: Your Choice
6𝑥−5𝑧=−11 𝑥−𝑦=−12 −4𝑥−4𝑦+5𝑧=−25
Answer

29. Solve the system by: Your Choice
6𝑥−5𝑧=−11 𝑥−𝑦=−12 −4𝑥−4𝑦+5𝑧=−25
𝑥−𝑦=−12
𝑥−6=−12
𝑥=−6
2𝑥−4𝑦=−36
−𝟐(𝑥−𝑦=−12)
−2𝑥+2𝑦=24
−2𝑦=−12
𝑦=6
6𝑥−5𝑧=−11
−4𝑥−4𝑦+5𝑧=−25
2𝑥−4𝑦=−36
6𝑥−5𝑧=−11
6 −6 −5𝑧=−11
−36−5𝑧=−11
−5𝑧=25
𝑧=−5
6 0 −5 1 −1 0 −4 − −11 −12 −25
(−6, 6, −5)

30. Solve the system by: Your Choice
6𝑥−5𝑦+𝑧=−17 2𝑥−𝑦+𝑧=−5 𝑧=−3𝑥−8
Answer

31. Solve the system by: Your Choice
6𝑥−5𝑦+𝑧=−17 2𝑥−𝑦+𝑧=−5 𝑧=−3𝑥−8
6𝑥−5𝑦+ −3𝑥−8 =−17
3𝑥−5𝑦=−9
2𝑥−𝑦+ −3𝑥−8 =−5
−𝑥−𝑦=3
−𝑥−𝑦=3
−𝑥−(0)=3
−𝑥=3
𝑥=3
3𝑥−5𝑦=−9
𝟑(−𝑥−𝑦=3)
−3𝑥−3𝑦=9
−8𝑦=0
𝑦=0
6 −5 1 2 − −17 −5 −8
𝑧=−3𝑥−8
𝑧=−3 3 −8
𝑧=1
(3, 0, 1)

32. Solve the system by: Your Choice
𝑥+6𝑦−2𝑧=25 𝑥−5𝑦−3𝑧=9 6𝑥+𝑦+6𝑧=−28
Answer

33. Solve the system 𝑥+6𝑦−2𝑧=25 −𝟏(𝑥−5𝑦−3𝑧=9) −𝑥+5𝑦+3𝑧=−9 11𝑦+𝑧=16
6𝑥+𝑦+6𝑧=−28
−𝟔(𝑥−5𝑦−3𝑧=9)
−6𝑥+30𝑦+18𝑧=−54
31𝑦+24𝑧=−82
𝑥+6𝑦−2𝑧=25 𝑥−5𝑦−3𝑧=9 6𝑥+𝑦+6𝑧=−28
31𝑦+24𝑧=−82
−𝟐𝟒(11𝑦+𝑧=16)
−264𝑦−24𝑧=−384
−233𝑦=−466
𝑦=2
11𝑦+𝑧=16
11 2 +𝑧=16
22+𝑧=16
𝑧=−6
𝑥−5𝑦−3𝑧=9
𝑥−5 2 −3 −6 =9
𝑥−10+18=9
𝑥=1
1 6 −2 1 −5 − −28
(1, 2, −6)

34. Graph the solution to each inequality
7𝑥−3𝑦>−9
Answer

35. Graph the solution to each inequality
7𝑥−3𝑦>−9

36. Graph the solution to each inequality
𝑥−𝑦≤2 𝑦<4𝑥+1
Answer

37. Graph the solution to each inequality
𝑥−𝑦≤2 𝑦<4𝑥+1

38. Graph the solution to each inequality
𝑦≥2 𝑥−3 −4 𝑦<− 2 3 𝑥+1
Answer

39. Graph the solution to each inequality
𝑦≥2 𝑥−3 −4 𝑦<− 2 3 𝑥+1

40. Graph the feasible region and find the point that maximize the function: 3𝑥−4𝑦
𝑦≥𝑥−2 𝑥+2𝑦≤8 𝑦≥1 𝑥≥0
Answer

41. Graph the feasible region and find the point that maximizes the function: 3𝑥−4𝑦
𝑦≥𝑥−2 𝑥+2𝑦≤8 𝑦≥1 𝑥≥0
(0, 1)
3 0 −4 1 =−4
(0, 5)
3 0 −4 5 =−20
(4, 2)
3 4 −4 2 =4
(3, 1)
3 3 −4 1 =5

42. Graph the feasible region and find the point that maximizes the function: 4𝑥+5𝑦
−𝑥+2𝑦≤4 𝑥+𝑦≤8 𝑦≥1 𝑥≥2
Answer

43. Graph the feasible region and find the point that maximizes the function: 4𝑥+5𝑦
−𝑥+2𝑦≤4 𝑥+𝑦≤8 𝑦≥1 𝑥≥2
(2, 1)
=13
(2, 2)
=18
(5, 4)
=40
(7, 1)
=33

44. The school that Danielle goes to is selling tickets to a choral performance. On the first day of ticket sales the school sold 7 senior citizen tickets and 3 child tickets for a total of $74. The school took in $135 on the second day by selling 14 senior citizen tickets and 5 child tickets. Write a system of equations to find the price of one senior citizen ticket and one child ticket?
Answer

45. 𝑆=Senior Citizen Tickets 𝐶=Child Tickets 7𝑆+3𝐶=74 14𝑆+5𝐶=135
The school that Danielle goes to is selling tickets to a choral performance. On the first day of ticket sales the school sold 7 senior citizen tickets and 3 child tickets for a total of $74. The school took in $135 on the second day by selling 14 senior citizen tickets and 5 child tickets. Write a system of equations to find the price of one senior citizen ticket and one child ticket?
𝑆=Senior Citizen Tickets
𝐶=Child Tickets
7𝑆+3𝐶=74 14𝑆+5𝐶=135

46. Amanda and Eduardo each improved their yards by planting grass sod and shrubs. They bought their supplies from the same store. Amanda spent $83 on 7 ft² of grass sod and 3 shrubs. Eduardo spent $118 on 8 ft² of grass sod and 6 shrubs. Find the cost of one ft² of grass sod and the cost of one shrub.
Write a system of equation to solve the problem.
Answer

47. 𝐺=Grass Sod 𝑆=Shrubs 7𝐺+3𝑆=83 8𝐺+6𝑆=118
Amanda and Eduardo each improved their yards by planting grass sod and shrubs. They bought their supplies from the same store. Amanda spent $83 on 7 ft² of grass sod and 3 shrubs. Eduardo spent $118 on 8 ft² of grass sod and 6 shrubs. Find the cost of one ft² of grass sod and the cost of one shrub.
Write a system of equation to solve the problem.
𝐺=Grass Sod
𝑆=Shrubs
7𝐺+3𝑆=83 8𝐺+6𝑆=118

48. Meg has three dogs – Skippy, Gizmo, and Chopper
Meg has three dogs – Skippy, Gizmo, and Chopper. The sum of the dogs’ weights is 148 pounds. If you add three times Skippy’s weight to Gizmo’s weight, the sum is 8 pounds less than Chopper’s weight. If you subtract one-third of Skippy’s weight from four times Gizmo’s weight, the result is equal to twice Chopper’s weight.
Write a system of equations to find out how much each dog weighs? Then solve the system.
Answer

49. 𝑆=Skippy 𝐺=Gizmo 𝐶=Chopper 𝑆+𝐺+𝐶=148 3𝑆+𝐺=𝐶−8 4𝐺− 1 3 𝑆=2𝐶
Meg has three dogs – Skippy, Gizmo, and Chopper. The sum of the dogs’ weights is 148 pounds. If you add three times Skippy’s weight to Gizmo’s weight, the sum is 8 pounds less than Chopper’s weight. If you subtract one-third of Skippy’s weight from four times Gizmo’s weight, the result is equal to twice Chopper’s weight.
Write a system of equations to find out how much each dog weighs? Then solve the system.
𝑆=Skippy
𝐺=Gizmo
𝐶=Chopper
𝑆+𝐺+𝐶=148 3𝑆+𝐺=𝐶−8 4𝐺− 1 3 𝑆=2𝐶
Skippy: 12 pounds
Gizmo: 46 pounds
Chopper: 90 pounds

50. You manage a health food store and budget $80 to buy ingredients to make 30 pounds of trail mix. Peanuts cost $2.50 per pound, raisins cost $2.00 per pound and granola cost $4.00 per pound. If you use twice as many pounds of peanuts as raisins, how many pounds of each ingredient should you buy?
Answer

51. 2.5𝑝+2𝑟+4𝑔=80 𝑝+𝑟+𝑔=30 𝑝=2𝑟 Peanuts (p): 16 pounds
You manage a health food store and budget $80 to buy ingredients to make 30 pounds of trail mix. Peanuts cost $2.50 per pound, raisins cost $2.00 per pound and granola cost $4.00 per pound. If you use twice as many pounds of peanuts as raisins, how many pounds of each ingredient should you buy?
2.5𝑝+2𝑟+4𝑔=80 𝑝+𝑟+𝑔=30 𝑝=2𝑟
Peanuts (p): 16 pounds
Raisins (r): 8 pounds
Granola (g): 6 pounds

52. You are making your summer movie plans and are working with following constraints:
It costs $8 to go to the movies at night.
It costs $5 to go to a matinee.
You want to go to at least as many night shows as matinees.
You want to spend at most $42
What is the greatest number of movies you can see?
Answer

53. 8𝑛+5𝑚≤42 𝑛≥𝑚 𝑛≥0 𝑚≥0 6 Movies: 3 night, 3 matinee 4 night, 2 matinee
You are making your summer movie plans and are working with following constraints:
It costs $8 to go to the movies at night.
It costs $5 to go to a matinee.
You want to go to at least as many night shows as matinees.
You want to spend at most $42
What is the greatest number of movies you can see?
8𝑛+5𝑚≤42 𝑛≥𝑚 𝑛≥0 𝑚≥0
6 Movies:
3 night, 3 matinee
4 night, 2 matinee