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**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

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**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

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**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

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**Without graphing, classify each system**

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

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**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

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**Without graphing, classify each system**

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

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**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

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**Without graphing, classify each system**

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

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**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

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**Without graphing, classify each system**

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

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**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

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**Without graphing, classify each system**

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

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**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

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**Solve the system by: Graphing**

𝑦=2𝑥−4 𝑥−4𝑦=−3 Answer

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**Solve the system by: Graphing**

𝑦=2𝑥−4 𝑥−4𝑦=−12 (4, 4)

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**Solve the system by: Substitution**

−2𝑥−𝑦=−9 𝑦=−5𝑥+15 Answer

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**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)

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**Solve the system by: Elimination**

−8𝑥−7𝑦=−28 5𝑥+6𝑦=24 Answer

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**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)

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**Solve the system by: Your Choice**

6𝑦+11+𝑥=0 8𝑥=−4−6𝑦 Answer

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**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)

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**Solve the system by: Your Choice**

−4𝑦=8𝑥+12 0=18𝑦+12𝑥−90 Answer

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**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)

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**Write a matrix to represent the system.**

6𝑥+2𝑦+𝑧=30 −3𝑥+3𝑧=0 −2𝑥+5𝑦+4𝑧=3 Answer

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**Write a matrix to represent the system.**

6𝑥+2𝑦+𝑧=30 −3𝑥+3𝑧=0 −2𝑥+5𝑦+4𝑧=3 6 2 1 −3 0 3 −

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**What system is represented by the matrix:**

− Answer

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**What system is represented by the matrix:**

− 2𝑥+7𝑦+𝑧=9 3𝑥−2𝑦=6 𝑥+2𝑦+𝑧=0

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**Solve the system by: Your Choice**

6𝑥−5𝑧=−11 𝑥−𝑦=−12 −4𝑥−4𝑦+5𝑧=−25 Answer

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**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)

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**Solve the system by: Your Choice**

6𝑥−5𝑦+𝑧=−17 2𝑥−𝑦+𝑧=−5 𝑧=−3𝑥−8 Answer

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**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)

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**Solve the system by: Your Choice**

𝑥+6𝑦−2𝑧=25 𝑥−5𝑦−3𝑧=9 6𝑥+𝑦+6𝑧=−28 Answer

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**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)

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**Graph the solution to each inequality**

7𝑥−3𝑦>−9 Answer

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**Graph the solution to each inequality**

7𝑥−3𝑦>−9

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**Graph the solution to each inequality**

𝑥−𝑦≤2 𝑦<4𝑥+1 Answer

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**Graph the solution to each inequality**

𝑥−𝑦≤2 𝑦<4𝑥+1

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**Graph the solution to each inequality**

𝑦≥2 𝑥−3 −4 𝑦<− 2 3 𝑥+1 Answer

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**Graph the solution to each inequality**

𝑦≥2 𝑥−3 −4 𝑦<− 2 3 𝑥+1

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**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

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**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

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**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

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**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

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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

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**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

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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

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**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

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**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

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**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

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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

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**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

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**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

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**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

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