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Linear System of Equations• Classify Systems• Independent• Dependent• Inconsistent
Methods for Solving• Tables• Graphing• Substitution• Elimination• Matrices
-5 5
-5
5
x
y
{𝑦=𝑥+3𝑥+𝑦=1
−2 1 3
−1 2 2
0 3 1[−𝟏 𝟏𝟏 𝟏
∨𝟑𝟏 ] [𝟏 𝟎
𝟎 𝟏|−𝟏𝟐 ]
(−1 ,2)
Linear System of Inequalities
{−𝒙+𝟐 𝒚 ≤𝟒𝒙+𝒚<𝟖𝒙 ≥𝟐𝒚 ≥𝟏
-1 10-1
10
x
y
-1 10-1
10
x
y
-1 10-1
10
x
y
x
y
x
yMaximize:
Pg. 187 #1 – 13
Linear SystemsClassifying
Systems
Solving 2 Variable Systems
Solving 3 Variable Systems
Inequalities Modeling(Application)
10 10 10 10 10
20 20 20 20 20
30 30 30 30 30
40 40 40 40 40
50 50 50 50 50
Answer
Without graphing, classify each system. (independent, dependent, inconsistent)
{ 5𝑥+3 𝑦=9
𝑦=− 35𝑥+3
Without graphing, classify each system. (independent, dependent, inconsistent)
{ 5𝑥+3 𝑦=9
𝑦=− 35𝑥+3
Independent
Answer
Without graphing, classify each system. (independent, dependent, inconsistent)
{12𝑥−4 𝑦=20𝑦=3𝑥+3
Without graphing, classify each system. (independent, dependent, inconsistent)
{12𝑥−4 𝑦=20𝑦=3𝑥+3
Inconsistent
Answer
Without graphing, classify each system. (independent, dependent, inconsistent)
{2𝑥+6 𝑦=12
𝑦=− 13𝑥+2
Without graphing, classify each system. (independent, dependent, inconsistent)
{2𝑥+6 𝑦=12
𝑦=− 13𝑥+2
Dependent
Answer
Without graphing, classify each system. (independent, dependent, inconsistent)
{−12 𝑥+2 𝑦=−1518 𝑥−3 𝑦=27
Without graphing, classify each system. (independent, dependent, inconsistent)
{−12 𝑥+2 𝑦=−1518 𝑥−3 𝑦=27
Inconsistent
Answer
Without graphing, classify each system. (independent, dependent, inconsistent)
{15 𝑥+6 𝑦=610 𝑥+4 𝑦=4
Without graphing, classify each system. (independent, dependent, inconsistent)
{15 𝑥+6 𝑦=610 𝑥+4 𝑦=4
Dependent
-5 5
-5
5
x
y
Answer
Solve the system by: Graphing
{ 𝑦=2 𝑥−4𝑥−4 𝑦=−3
Solve the system by: Graphing
{ 𝑦=2 𝑥−4𝑥−4 𝑦=−12
-5 5
-5
5
x
y
(4 , 4)
Answer
Solve the system by: Substitution
{−2 𝑥− 𝑦=−9𝑦=−5 𝑥+15
Solve the system by: Substitution
{−2 𝑥− 𝑦=−9𝑦=−5 𝑥+15
(2 ,5)
Answer
Solve the system by: Elimination
{−8 𝑥−7 𝑦=−285 𝑥+6 𝑦=24
Solve the system by: Elimination
{−8 𝑥−7 𝑦=−285 𝑥+6 𝑦=24
(0 ,4)
Answer
Solve the system by: Your Choice
{6 𝑦+11+𝑥=08 𝑥=−4−6 𝑦
Solve the system by: Your Choice
{6 𝑦+11+𝑥=08 𝑥=−4−6 𝑦
(1 ,−2)
Answer
Solve the system by: Your Choice
{ −4 𝑦=8 𝑥+120=18 𝑦+12𝑥−90
Solve the system by: Your Choice
{ −4 𝑦=8 𝑥+120=18 𝑦+12𝑥−90
(−6 ,9)
Answer
Write a matrix to represent the system.
{ 6 𝑥+2 𝑦+𝑧=30−3𝑥+3 𝑧=0
−2 𝑥+5 𝑦+4 𝑧=3
Write a matrix to represent the system.
{ 6 𝑥+2 𝑦+𝑧=30−3𝑥+3 𝑧=0
−2 𝑥+5 𝑦+4 𝑧=3
[ 6 2 1−3 0 3−2 5 4|
3 003 ]
Answer
What system is represented by the matrix:
[2 7 13 −2 01 2 1|
960 ]
What system is represented by the matrix:
{2𝑥+7 𝑦+𝑧=93 𝑥−2 𝑦=6𝑥+2 𝑦+𝑧=0
[2 7 13 −2 01 2 1|
960 ]
Answer
Solve the system by: Your Choice
{ 6 𝑥−5 𝑧=−11𝑥− 𝑦=−12
−4 𝑥−4 𝑦+5 𝑧=−25
Solve the system by: Your Choice
{ 6 𝑥−5 𝑧=−11𝑥− 𝑦=−12
−4 𝑥−4 𝑦+5 𝑧=−25
[ 6 0 −51 −1 0−4 −4 5 |−11
−12−25 ] (−6 ,6 ,−5)
Answer
Solve the system by: Your Choice
{6 𝑥−5 𝑦+𝑧=−172 𝑥− 𝑦+𝑧=−5𝑧=−3𝑥−8
Solve the system by: Your Choice
{6 𝑥−5 𝑦+𝑧=−172 𝑥− 𝑦+𝑧=−5𝑧=−3𝑥−8
[6 −5 12 −1 13 0 1|
−17−5−8 ]
(3 ,0 ,1)
Answer
Solve the system by: Your Choice
{ 𝑥+6 𝑦−2 𝑧=25𝑥−5 𝑦−3 𝑧=9
6 𝑥+ 𝑦+6 𝑧=−28
Solve the system
{ 𝑥+6 𝑦−2 𝑧=25𝑥−5 𝑦−3 𝑧=9
6 𝑥+ 𝑦+6 𝑧=−28
[1 6 −21 −5 −36 1 6 | 25
9−28 ] (1 ,2 ,−6)
-5 5
-5
5
x
y
Answer
Graph the solution to each inequality
7 𝑥−3 𝑦>−9
-5 5
-5
5
x
y
Graph the solution to each inequality
7 𝑥−3 𝑦>−9
-5 5
-5
5
x
y
Answer
Graph the solution to each inequality
{ 𝑥− 𝑦 ≤2𝑦<4 𝑥+1
-5 5
-5
5
x
y
Graph the solution to each inequality
{ 𝑥− 𝑦 ≤2𝑦<4 𝑥+1
Answer
Graph the solution to each inequality
{𝑦 ≥2|𝑥−3|−4
𝑦<− 23𝑥+1
-5 5
-5
5
x
y
-5 5
-5
5
x
y
Graph the solution to each inequality
{𝑦 ≥2|𝑥−3|−4
𝑦<− 23𝑥+1
-5 5
-5
5
x
y
Answer
Graph the feasible region and find the point that maximize the function:
{ 𝑦 ≥ 𝑥−2𝑥+2 𝑦 ≤8𝑦 ≥1𝑥≥0
-5 5
-5
5
x
y
-5 5
-5
5
x
y
Graph the feasible region and find the point that maximizes the function:
{ 𝑦 ≥ 𝑥−2𝑥+2 𝑦 ≤8𝑦 ≥1𝑥≥0
(0, 1)
(0, 5)
(4, 2)
(3, 1)
Answer
Graph the feasible region and find the point that maximizes the function:
{−𝑥+2 𝑦 ≤4𝑥+𝑦 ≤8𝑦 ≥1𝑥≥2
-1 10-1
10
x
y
-1 10-1
10
x
y
Graph the feasible region and find the point that maximizes the function:
{−𝑥+2 𝑦 ≤4𝑥+𝑦 ≤8𝑦 ≥1𝑥≥2
-1 10-1
10
x
y
(2, 1)
(2, 2)
(5, 4)
(7, 1)
Answer
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?
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
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.
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
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.
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.
Skippy: 12 poundsGizmo: 46 poundsChopper: 90 pounds
Answer
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?
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=30𝑝=2𝑟
Peanuts (p): 16 poundsRaisins (r): 8 poundsGranola (g): 6 pounds
Answer
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 $42What is the greatest number of movies you can see?
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 $42What is the greatest number of movies you can see?
{8𝑛+5𝑚≤ 42𝑛≥𝑚𝑛≥0𝑚≥0
1 2 3 4 5 6 7
1
2
3
4
5
6
7
Matinee
Night
6 Movies:3 night, 3 matinee4 night, 2 matinee