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Linear System of Equations

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1 Linear System of Equations
Classify Systems Independent Dependent Inconsistent Methods for Solving Tables Graphing Substitution Elimination Matrices 𝑦=π‘₯+3 π‘₯+𝑦=1 (βˆ’1, 2) 𝒙+(𝒙+πŸ‘)=𝟏 πŸπ’™=βˆ’πŸ 𝒙=βˆ’πŸ π’š= βˆ’πŸ +πŸ‘=𝟐 βˆ’π’™+π’š=πŸ‘ 𝒙+π’š=𝟏 πŸπ’š=πŸ’ π’š=𝟐 𝟐=𝒙+πŸ‘ βˆ’πŸ=𝒙 π‘₯ 𝑦=π‘₯+3 π‘₯+𝑦=1 βˆ’2 1 3 βˆ’1 2 βˆ’πŸ 𝟏 𝟏 𝟏 | πŸ‘ 𝟏 𝟏 𝟎 𝟎 𝟏 βˆ’πŸ 𝟐

2 Linear System of Inequalities
Maximize: 3π‘₯+2𝑦 βˆ’π’™+πŸπ’šβ‰€πŸ’ 𝒙+π’š<πŸ– 𝒙β‰₯𝟐 π’šβ‰₯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) 5π‘₯+3𝑦=9 𝑦=βˆ’ 3 5 π‘₯+3 Answer

5 Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent) 5π‘₯+3𝑦=9 3𝑦=βˆ’5π‘₯+9 𝑦=βˆ’ 5 3 π‘₯+3 5π‘₯+3𝑦=9 𝑦=βˆ’ 3 5 π‘₯+3 Independent

6 Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent) 12π‘₯βˆ’4𝑦=20 𝑦=3π‘₯+3 Answer

7 Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent) 12π‘₯βˆ’4𝑦=20 βˆ’4𝑦=βˆ’12π‘₯+20 𝑦=3π‘₯βˆ’5 12π‘₯βˆ’4𝑦=20 𝑦=3π‘₯+3 Inconsistent

8 Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent) 2π‘₯+6𝑦=12 𝑦=βˆ’ 1 3 π‘₯+2 Answer

9 Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent) 2π‘₯+6𝑦=12 𝑦=βˆ’ 1 3 π‘₯+2 2π‘₯+6𝑦=12 6𝑦=βˆ’2π‘₯+12 𝑦=βˆ’ 1 3 π‘₯+2 Dependent

10 Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent) βˆ’12π‘₯+2𝑦=βˆ’15 18π‘₯βˆ’3𝑦=27 Answer

11 Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent) βˆ’12π‘₯+2𝑦=βˆ’15 18π‘₯βˆ’3𝑦=27 βˆ’12π‘₯+2𝑦=βˆ’15 2𝑦=12π‘₯βˆ’15 𝑦=6π‘₯βˆ’ 15 2 18π‘₯βˆ’3𝑦=27 βˆ’3𝑦=βˆ’18π‘₯+27 𝑦=6π‘₯βˆ’9 Inconsistent

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

13 Without graphing, classify each system
Without graphing, classify each system. (independent, dependent, inconsistent) 15π‘₯+6𝑦=6 10π‘₯+4𝑦=4 15π‘₯+6𝑦=6 6𝑦=βˆ’15π‘₯+6 𝑦=βˆ’ 5 2 π‘₯+1 10π‘₯+4𝑦=4 4𝑦=βˆ’10π‘₯+4 𝑦=βˆ’ 5 2 π‘₯+1 Dependent

14 Solve the system by: Graphing
𝑦=2π‘₯βˆ’4 π‘₯βˆ’4𝑦=βˆ’3 Answer

15 Solve the system by: Graphing
𝑦=2π‘₯βˆ’4 π‘₯βˆ’4𝑦=βˆ’12 (4, 4)

16 Solve the system by: Substitution
βˆ’2π‘₯βˆ’π‘¦=βˆ’9 𝑦=βˆ’5π‘₯+15 Answer

17 Solve the system by: Substitution
βˆ’2π‘₯βˆ’π‘¦=βˆ’9 𝑦=βˆ’5π‘₯+15 βˆ’2π‘₯βˆ’π‘¦=βˆ’9 βˆ’2π‘₯βˆ’(βˆ’5π‘₯+15)=βˆ’9 βˆ’2π‘₯+5π‘₯βˆ’15=βˆ’9 3π‘₯=6 π‘₯=2 𝑦=βˆ’5π‘₯+15 𝑦=βˆ’ 𝑦=5 (2, 5)

18 Solve the system by: Elimination
βˆ’8π‘₯βˆ’7𝑦=βˆ’28 5π‘₯+6𝑦=24 Answer

19 Solve the system by: Elimination
βˆ’8π‘₯βˆ’7𝑦=βˆ’28 5π‘₯+6𝑦=24 πŸ” βˆ’8π‘₯βˆ’7𝑦=βˆ’28 πŸ•(5π‘₯+6𝑦=24) βˆ’48π‘₯βˆ’42𝑦=βˆ’168 35π‘₯+42𝑦=168 βˆ’13π‘₯=0 π‘₯=0 5π‘₯+6𝑦=24 5 0 +6𝑦=24 6𝑦=24 𝑦=4 (0, 4)

20 Solve the system by: Your Choice
6𝑦+11+π‘₯=0 8π‘₯=βˆ’4βˆ’6𝑦 Answer

21 Solve the system by: Your Choice
6𝑦+11+π‘₯=0 8π‘₯=βˆ’4βˆ’6𝑦 π‘₯+6𝑦=βˆ’11 βˆ’8π‘₯βˆ’6𝑦=4 βˆ’7π‘₯=βˆ’7 π‘₯=1 6𝑦+11+π‘₯=0 6𝑦 =0 6𝑦=βˆ’12 𝑦=βˆ’2 (1, βˆ’2)

22 Solve the system by: Your Choice
βˆ’4𝑦=8π‘₯+12 0=18𝑦+12π‘₯βˆ’90 Answer

23 Solve the system by: Your Choice
βˆ’4𝑦=8π‘₯+12 0=18𝑦+12π‘₯βˆ’90 0=18𝑦+12π‘₯βˆ’90 0=18 βˆ’2π‘₯βˆ’3 +12π‘₯βˆ’90 0=βˆ’36π‘₯βˆ’54+12π‘₯βˆ’90 144=βˆ’24π‘₯ βˆ’6=π‘₯ βˆ’4𝑦=8π‘₯+12 𝑦=βˆ’2π‘₯βˆ’3 𝑦=βˆ’2 βˆ’6 βˆ’3 𝑦=9 (βˆ’6, 9)

24 Write a matrix to represent the system.
6π‘₯+2𝑦+𝑧=30 βˆ’3π‘₯+3𝑧=0 βˆ’2π‘₯+5𝑦+4𝑧=3 Answer

25 Write a matrix to represent the system.
6π‘₯+2𝑦+𝑧=30 βˆ’3π‘₯+3𝑧=0 βˆ’2π‘₯+5𝑦+4𝑧=3 6 2 1 βˆ’3 0 3 βˆ’

26 What system is represented by the matrix:
βˆ’ Answer

27 What system is represented by the matrix:
βˆ’ 2π‘₯+7𝑦+𝑧=9 3π‘₯βˆ’2𝑦=6 π‘₯+2𝑦+𝑧=0

28 Solve the system by: Your Choice
6π‘₯βˆ’5𝑧=βˆ’11 π‘₯βˆ’π‘¦=βˆ’12 βˆ’4π‘₯βˆ’4𝑦+5𝑧=βˆ’25 Answer

29 Solve the system by: Your Choice
6π‘₯βˆ’5𝑧=βˆ’11 π‘₯βˆ’π‘¦=βˆ’12 βˆ’4π‘₯βˆ’4𝑦+5𝑧=βˆ’25 π‘₯βˆ’π‘¦=βˆ’12 π‘₯βˆ’6=βˆ’12 π‘₯=βˆ’6 2π‘₯βˆ’4𝑦=βˆ’36 βˆ’πŸ(π‘₯βˆ’π‘¦=βˆ’12) βˆ’2π‘₯+2𝑦=24 βˆ’2𝑦=βˆ’12 𝑦=6 6π‘₯βˆ’5𝑧=βˆ’11 βˆ’4π‘₯βˆ’4𝑦+5𝑧=βˆ’25 2π‘₯βˆ’4𝑦=βˆ’36 6π‘₯βˆ’5𝑧=βˆ’11 6 βˆ’6 βˆ’5𝑧=βˆ’11 βˆ’36βˆ’5𝑧=βˆ’11 βˆ’5𝑧=25 𝑧=βˆ’5 6 0 βˆ’5 1 βˆ’1 0 βˆ’4 βˆ’ βˆ’11 βˆ’12 βˆ’25 (βˆ’6, 6, βˆ’5)

30 Solve the system by: Your Choice
6π‘₯βˆ’5𝑦+𝑧=βˆ’17 2π‘₯βˆ’π‘¦+𝑧=βˆ’5 𝑧=βˆ’3π‘₯βˆ’8 Answer

31 Solve the system by: Your Choice
6π‘₯βˆ’5𝑦+𝑧=βˆ’17 2π‘₯βˆ’π‘¦+𝑧=βˆ’5 𝑧=βˆ’3π‘₯βˆ’8 6π‘₯βˆ’5𝑦+ βˆ’3π‘₯βˆ’8 =βˆ’17 3π‘₯βˆ’5𝑦=βˆ’9 2π‘₯βˆ’π‘¦+ βˆ’3π‘₯βˆ’8 =βˆ’5 βˆ’π‘₯βˆ’π‘¦=3 βˆ’π‘₯βˆ’π‘¦=3 βˆ’π‘₯βˆ’(0)=3 βˆ’π‘₯=3 π‘₯=3 3π‘₯βˆ’5𝑦=βˆ’9 πŸ‘(βˆ’π‘₯βˆ’π‘¦=3) βˆ’3π‘₯βˆ’3𝑦=9 βˆ’8𝑦=0 𝑦=0 6 βˆ’5 1 2 βˆ’ βˆ’17 βˆ’5 βˆ’8 𝑧=βˆ’3π‘₯βˆ’8 𝑧=βˆ’3 3 βˆ’8 𝑧=1 (3, 0, 1)

32 Solve the system by: Your Choice
π‘₯+6π‘¦βˆ’2𝑧=25 π‘₯βˆ’5π‘¦βˆ’3𝑧=9 6π‘₯+𝑦+6𝑧=βˆ’28 Answer

33 Solve the system π‘₯+6π‘¦βˆ’2𝑧=25 βˆ’πŸ(π‘₯βˆ’5π‘¦βˆ’3𝑧=9) βˆ’π‘₯+5𝑦+3𝑧=βˆ’9 11𝑦+𝑧=16
6π‘₯+𝑦+6𝑧=βˆ’28 βˆ’πŸ”(π‘₯βˆ’5π‘¦βˆ’3𝑧=9) βˆ’6π‘₯+30𝑦+18𝑧=βˆ’54 31𝑦+24𝑧=βˆ’82 π‘₯+6π‘¦βˆ’2𝑧=25 π‘₯βˆ’5π‘¦βˆ’3𝑧=9 6π‘₯+𝑦+6𝑧=βˆ’28 31𝑦+24𝑧=βˆ’82 βˆ’πŸπŸ’(11𝑦+𝑧=16) βˆ’264π‘¦βˆ’24𝑧=βˆ’384 βˆ’233𝑦=βˆ’466 𝑦=2 11𝑦+𝑧=16 11 2 +𝑧=16 22+𝑧=16 𝑧=βˆ’6 π‘₯βˆ’5π‘¦βˆ’3𝑧=9 π‘₯βˆ’5 2 βˆ’3 βˆ’6 =9 π‘₯βˆ’10+18=9 π‘₯=1 1 6 βˆ’2 1 βˆ’5 βˆ’ βˆ’28 (1, 2, βˆ’6)

34 Graph the solution to each inequality
7π‘₯βˆ’3𝑦>βˆ’9 Answer

35 Graph the solution to each inequality
7π‘₯βˆ’3𝑦>βˆ’9

36 Graph the solution to each inequality
π‘₯βˆ’π‘¦β‰€2 𝑦<4π‘₯+1 Answer

37 Graph the solution to each inequality
π‘₯βˆ’π‘¦β‰€2 𝑦<4π‘₯+1

38 Graph the solution to each inequality
𝑦β‰₯2 π‘₯βˆ’3 βˆ’4 𝑦<βˆ’ 2 3 π‘₯+1 Answer

39 Graph the solution to each inequality
𝑦β‰₯2 π‘₯βˆ’3 βˆ’4 𝑦<βˆ’ 2 3 π‘₯+1

40 Graph the feasible region and find the point that maximize the function: 3π‘₯βˆ’4𝑦
𝑦β‰₯π‘₯βˆ’2 π‘₯+2𝑦≀8 𝑦β‰₯1 π‘₯β‰₯0 Answer

41 Graph the feasible region and find the point that maximizes the function: 3π‘₯βˆ’4𝑦
𝑦β‰₯π‘₯βˆ’2 π‘₯+2𝑦≀8 𝑦β‰₯1 π‘₯β‰₯0 (0, 1) 3 0 βˆ’4 1 =βˆ’4 (0, 5) 3 0 βˆ’4 5 =βˆ’20 (4, 2) 3 4 βˆ’4 2 =4 (3, 1) 3 3 βˆ’4 1 =5

42 Graph the feasible region and find the point that maximizes the function: 4π‘₯+5𝑦
βˆ’π‘₯+2𝑦≀4 π‘₯+𝑦≀8 𝑦β‰₯1 π‘₯β‰₯2 Answer

43 Graph the feasible region and find the point that maximizes the function: 4π‘₯+5𝑦
βˆ’π‘₯+2𝑦≀4 π‘₯+𝑦≀8 𝑦β‰₯1 π‘₯β‰₯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 𝐢=Child Tickets 7𝑆+3𝐢=74 14𝑆+5𝐢=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 𝐢=Child Tickets 7𝑆+3𝐢=74 14𝑆+5𝐢=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 ftΒ² of grass sod and 3 shrubs. Eduardo spent $118 on 8 ftΒ² of grass sod and 6 shrubs. Find the cost of one ftΒ² 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 ftΒ² of grass sod and 3 shrubs. Eduardo spent $118 on 8 ftΒ² of grass sod and 6 shrubs. Find the cost of one ftΒ² 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 𝐢=Chopper 𝑆+𝐺+𝐢=148 3𝑆+𝐺=πΆβˆ’8 4πΊβˆ’ 1 3 𝑆=2𝐢
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 𝐢=Chopper 𝑆+𝐺+𝐢=148 3𝑆+𝐺=πΆβˆ’8 4πΊβˆ’ 1 3 𝑆=2𝐢 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+2π‘Ÿ+4𝑔=80 𝑝+π‘Ÿ+𝑔=30 𝑝=2π‘Ÿ 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+2π‘Ÿ+4𝑔=80 𝑝+π‘Ÿ+𝑔=30 𝑝=2π‘Ÿ 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+5π‘šβ‰€42 𝑛β‰₯π‘š 𝑛β‰₯0 π‘šβ‰₯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+5π‘šβ‰€42 𝑛β‰₯π‘š 𝑛β‰₯0 π‘šβ‰₯0 6 Movies: 3 night, 3 matinee 4 night, 2 matinee


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