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Section 8.1: Systems of Linear Equations, Substitution and Elimination
#1-6: Verify that the given values are solutions to the system of equations.
1)
3 x  2 y  11
(3,1)
2x  y  5
2)
4 x  3 y  10
(4,2)
2 x  y  10
2
1
x y 3
3) 3
(3, 2)
2
2x  y  4
4
3
x  y 1
4) 5
(5,7)
7
2 x  y  17
x yz 2
5) 2 x  3 y  2 z  2 (3,2,1)
2x  y  2z  2
6) 3 x  y  z  3 (2,4,1)
x  y  2z  3
x yz 7
#7 – 20: Solve each system of equations using the substitution method. If the system has no solutions
say, that it is inconsistent. If the system has infinitely many solutions write your answer in the form
{(𝑥, 𝑦)|𝑦 = 𝑚𝑥 + 𝑏}
7) 𝑦 = 2𝑥 − 4
−2𝑥 + 3𝑦 = 0
8)
x  2y  3
4x  y  5
9)
3 x  2 y  11
2x  y  5
10)
4x  3y  5
2x  y  5
2
1
x y 3
11) 3
2
2x  y  4
2
1
x y 3
12) 5
2
2x  y  8
4
1
x y 5
13) 3
5
x  y  2
4
1
x y 5
14) 7
3
x y 4
15)
x  2y  7
2 x  4 y  3
16)
x  3y  7
5 x  15 y  9
Section 8.1: Systems of Linear Equations, Substitution and Elimination
17)
4x  2 y  6
12 x  6 y  15
18)
6x  2 y  3
12 x  4 y  15
19)
4x  2 y  6
12 x  6 y  18
20)
6x  2 y  3
12 x  4 y  6
#21 – 36: Solve each system of equations using the elimination method. If the system has no solutions
say, that it is inconsistent. If the system has infinitely many solutions write your answer in the form
{(𝑥, 𝑦)|𝑦 = 𝑚𝑥 + 𝑏}
21)
3 x  2 y  11
2x  y  5
22)
4 x  3 y  10
2 x  y  10
23)
4x  2 y  7
2 x  5 y  3
24)
6 x  2 y  26
2 x  7 y  24
25)
5x  2 y  3
3 x  5 y  15
26)
4x  2 y  3
3x  5 y  1
2
1
x y 2
27) 4
2
x y 4
2
1
x y 3
28) 3
2
2x  3y  8
4
1
x y 5
29) 3
5
x  y  2
4
1
x y 5
30) 7
3
x y 4
31)
x  2y  7
5 x  10 y  3
32)
x  3y  7
3x  9 y  4
33)
4x  2 y  6
8 x  4 y  12
34)
6x  2 y  3
 18 x  6 y  9
35)
4x  2 y  6
2x  y  3
36)
x  2y  3
5 x  10 y  15
Section 8.1: Systems of Linear Equations, Substitution and Elimination
Solving systems with 3 equations and 3 unknowns.
Note: If at any time all of the variables cancel out and you are left with:
a) 0= some number that is not 0, like 0 = 6, you are done just write no solution.
b) 0=0 there are infinitely many solutions: do the following:



Find or create an equation with y and z, solve for y (
Find or create an equation with x and z, solve for x
Write solution (x,y,z) use the above bullets for the x and y
General solution strategy:
1) Pick an equation pair it with the other two. This creates two pairs of equations.
2) Pick a letter to drop (don’t pick the z) then drop the same letter from each pair. This will create two
equations with two unknowns.
3) Take the two equations created in the last step and solve them using the elimination method. This
will give answers for 2 of the 3 variables.
4) Substitute the answers from part 3 into one of the original equations and solve for the remaining
variable. Write your solution (x,y,z) but use numbers for the x, y and z.
#37-51: Solve each system of equations. If the system has no solutions say, that it is inconsistent. If the
system has infinitely many solutions write your answer in the form
{(𝑥, 𝑦, 𝑧)|𝑥 = 𝑎𝑧 + 𝑏, 𝑦 = 𝑐𝑧 + 𝑑, 𝑧}
(That is solve for x in terms of z, and also solve for y in terms of z).
Solve the system of linear equations and check your solutions.
37)
−𝑥 + 2𝑦 − 𝑧 = −17
2𝑥 − 𝑦 + 𝑧 = 21
3𝑥 + 2𝑦 + 𝑧 = 19
(pair the middle equation with the other 2 and drop out the y’s)
Section 8.1: Systems of Linear Equations, Substitution and Elimination
38)
3𝑥 − 2𝑦 + 𝑧 = 15
−𝑥 + 𝑦 + 2𝑧 = −10
𝑥 − 𝑦 − 4𝑧 = 14
(pair the bottom equation with the other 2 and drop out the z’s)
Answer (5, -1, -2)
39)
2𝑥 + 𝑦 − 5𝑧 = −11
−𝑥 + 𝑦 + 2𝑧 = 7
(pair the middle equation with the other 2 and drop out the x’s)
𝑥 − 3𝑦 + 𝑧 = −2
40)
−5𝑥 − 𝑦 + 3𝑧 = −14
−2𝑥 + 2𝑦 − 6𝑧 = 16
𝑥 + 7𝑦 + 2𝑧 = −5
41)
2𝑥 + 2𝑦 − 𝑧 = 2
𝑥 − 3𝑦 + 𝑧 = −28
−𝑥 + 𝑦 + 5𝑧 = 24
42)
−𝑥 + 𝑦 + 2𝑧 = 1
2𝑥 + 3𝑦 + 𝑧 = −2
5𝑥 + 4𝑦 + 2𝑧 = 4
43)
4𝑥 + 4𝑦 + 4𝑧 = 12
4𝑥 − 2𝑦 − 8𝑧 = −12
5𝑥 + 3𝑦 + 8𝑧 = 21
start anyway you like
Answer (1, 0, -3)
start anyway your like
start anyway you like
Answer (2, -3, 3)
start anyway you like
44)
2𝑥 − 3𝑦 − 𝑧 = 0
−𝑥 + 2𝑦 + 𝑧 = 5
3𝑥 − 4𝑦 − 𝑧 = 1
(pair the middle equation with the other 2 and drop out the z’s)
Answer (no solution)
Section 8.1: Systems of Linear Equations, Substitution and Elimination
45)
2𝑥 − 2𝑦 + 3𝑧 = 6
4𝑥 − 3𝑦 + 2𝑧 = 0
−2𝑥 + 3𝑦 − 7𝑧 = 1
46)
3𝑥 − 2𝑦 + 2𝑧 = 6
7𝑥 − 3𝑦 + 2𝑧 = −1
2𝑥 − 3𝑦 + 4𝑧 = 0
(pair the bottom equation with the other 2 and drop out the x’s)
(pair the top equation with the other 2 and drop out z’s)
Answer (no solution)
47)
2𝑥 + 𝑦 − 𝑧 = −2
𝑥 + 2𝑦 − 𝑧 = −9
𝑥 − 4𝑦 + 𝑧 = 1
48)
2𝑥 − 3𝑦 − 𝑧 = 0
3𝑥 + 2𝑦 + 2𝑧 = 2
𝑥 + 5𝑦 + 3𝑧 = 2
49)
𝑥−𝑦−𝑧 =1
−𝑥 + 2𝑦 − 3𝑧 = −4
3𝑥 − 2𝑦 − 7𝑧 = 0
(pair the bottom equation with the other 2 and drop out the z’s)
(pair the top equation with the other 2 and drop out the z’s)
Answer (infinite)
(pair the middle equation with the other 2 and drop out the y’s)
50) (answer in video is not correct answer should be {(𝑥, 𝑦, 𝑧)|𝑥 = −5𝑧 + 22, 𝑦 = −3𝑧 + 7} )
𝑥 − 2𝑦 − 𝑧 = 8
2𝑥 − 3𝑦 + 𝑧 = 23
(pair the top equation with the other 2 and drop out the z’s)
4𝑥 − 5𝑦 + 5𝑧 = 53
Answer (infinite)
51)
𝑥 + 2𝑦 − 𝑧 = 8
−𝑥 − 3𝑦 + 𝑧 = 23
𝑥 + 𝑦 + 5𝑧 = 39
(pair the middle equation with the other 2 and drop out the z’s)
Section 8.2: Systems of Linear Equations – Matrices
Solve the system of equations using matrices and row operations. Check your answer using the RREF
feature on your calculator.
1)
6 x  2 y  10
2x  y  5
2)
8 x  3 y  2
2 x  y  4
3)
4 x  3 y  2
x  5 y  9
4)
5 x  2 y  1.
x  7 y  10
5)
5 x  y  7
3x  2 y  12
6)
4 x  y  11
3x  5 y  37
7)
3 x  2 y  11
2x  y  5
8)
4 x  3 y  10
2 x  y  10
9)
4x  2 y  7
2 x  5 y  3
10)
6 x  2 y  26
2 x  7 y  24
12)
4x  2 y  3
3x  5 y  1
11)
5 x  2 y  22
3x  5 y  7
13)
−𝑥 + 𝑦 + 2𝑧 = 1
2𝑥 + 3𝑦 + 𝑧 = −2
5𝑥 + 4𝑦 + 2𝑧 = 4
14)
3𝑥 − 2𝑦 + 𝑧 = 15
−𝑥 + 𝑦 + 2𝑧 = −10
𝑥 − 𝑦 − 4𝑧 = 14
15)
−5𝑥 − 𝑦 + 3𝑧 = −14
−2𝑥 + 2𝑦 − 6𝑧 = 16
𝑥 + 7𝑦 + 2𝑧 = −5
16)
4𝑥 + 4𝑦 + 4𝑧 = 12
4𝑥 − 2𝑦 − 8𝑧 = −12
5𝑥 + 3𝑦 + 8𝑧 = 21
17)
−𝑥 + 2𝑦 − 𝑧 = −17
2𝑥 − 𝑦 + 𝑧 = 21
3𝑥 + 2𝑦 + 𝑧 = 19
18)
𝑥 + 𝑦 + 2𝑧 = 6
2𝑥 + 3𝑦 + 𝑧 = 11
5𝑥 + 4𝑦 + 2𝑧 = 19
19)
4𝑥 + 𝑦 + 𝑧 = 9
3𝑥 − 2𝑦 + 𝑧 = 4
5𝑥 − 4𝑦 + 𝑧 = 6
20)
𝑥−𝑦+𝑧 =2
2𝑥 + 𝑦 + 𝑧 = 5
7𝑥 + 4𝑦 − 𝑧 = 9
Section 8.3: Systems of Linear Equations – Determinants
#1 – 8: Find the value of the determinant of the following matrices
4 5

1 2 
  2 4
1
3
3  5 

0  2 
 4 5

 1 0 
1) 
2) 
3) 
4) 
3 5  1


5) 4 2 3


2 5 6 
1 0  1 


6) 7 2 4


2 3  5
 3 2 8 


7) 0 6  3


2 1 4 
3 2 0 


8) 4  1 3


1  5 6 
Solve the system of equations using Cramer’s rule. Check your answer using the RREF feature on your
calculator.
9)
6 x  2 y  10
2x  y  5
10)
8 x  3 y  2
2 x  y  4
11)
4 x  3 y  2
x  5 y  9
12)
5 x  2 y  1.
x  7 y  10
13)
5 x  y  7
3x  2 y  12
14)
4 x  y  11
3x  5 y  37
15)
3 x  2 y  11
2x  y  5
16)
4 x  3 y  10
2 x  y  10
17)
4x  2 y  7
2 x  5 y  3
18)
6 x  2 y  26
2 x  7 y  24
19)
5 x  2 y  22
3x  5 y  7
20)
4x  2 y  3
3x  5 y  1
21)
−𝑥 + 𝑦 + 2𝑧 = 1
2𝑥 + 3𝑦 + 𝑧 = −2
5𝑥 + 4𝑦 + 2𝑧 = 4
22)
3𝑥 − 2𝑦 + 𝑧 = 15
−𝑥 + 𝑦 + 2𝑧 = −10
𝑥 − 𝑦 − 4𝑧 = 14
Section 8.3: Systems of Linear Equations – Determinants
23)
−5𝑥 − 𝑦 + 3𝑧 = −14
−2𝑥 + 2𝑦 − 6𝑧 = 16
𝑥 + 7𝑦 + 2𝑧 = −5
24)
4𝑥 + 4𝑦 + 4𝑧 = 12
4𝑥 − 2𝑦 − 8𝑧 = −12
5𝑥 + 3𝑦 + 8𝑧 = 21
25)
−𝑥 + 2𝑦 − 𝑧 = −17
2𝑥 − 𝑦 + 𝑧 = 21
3𝑥 + 2𝑦 + 𝑧 = 19
26)
𝑥 + 𝑦 + 2𝑧 = 6
2𝑥 + 3𝑦 + 𝑧 = 11
5𝑥 + 4𝑦 + 2𝑧 = 19
27)
4𝑥 + 𝑦 + 𝑧 = 9
3𝑥 − 2𝑦 + 𝑧 = 4
5𝑥 − 4𝑦 + 𝑧 = 6
28)
𝑥−𝑦+𝑧 =2
2𝑥 + 𝑦 + 𝑧 = 5
7𝑥 + 4𝑦 − 𝑧 = 9
Section 8.4: Matrix Algebra
Use the following matrices to answer all the problems in this section.
  2 4
A= 
1
3
4 5
B= 

1 2 
3 2 0 


D = 4 1 3


1  5 6 
E= 
  3 2 2

5  1 3 
1 0  1 


C= 7 2 4


2 3  5
1  2


F= 5 2


6 7 
Perform the indicated operation
1) 3A
2) 5B
3) -2F
4) -4E
5) A + B
6) C + D
7) 2A+3B
8) 2C +6D
9) 3A – 2B
10) 5C – 4D
11) A + C
12) B + D
13) D + E
14) E + F
15) AB
16) BA
17) FA
18) EF
19) CD
20) DC
21) BC
22) AD
23) FE
24) AF
Section 8.5: systems on non-linear equations
Solve the following systems of equations. Be sure to check your answers.
1)
3)
5)
7)
9)
11)
4x  y  2
x3  2  y  0
 2 x  y  5
x 2  y 2  25
x2  y  0
8x 2  4 x  y  0
y  2 x 2  2
y  2x 4  4x 2  2
3 x  y  4
2)
4)
6)
8)
10)
x y0
2
x2  y  0
12)
x y 0
y  x
13)
15)
17)
19)
y  x  4x  2x
3
2
x  7 y  6  12
x  y  36
2
2
x  7 y  6  4
x  y  20
2
2
3x  y  8
x 2 y  6
2
14)
16)
18)
20)
x y 0
x 3  5x  y  0
3x  y  5
x2  y2  5
x  y2  0
y 3  5x  6 y  0
y  x 3  3x 2  4
y  2 x  4
x  2y  1
3x  y 2  8
x y 4
x 2  y 2  26
x y 9
x 3  5x  y  4
x  2y  0
x2  y2  3
x  2y  4
x2  y2  3
x y 3
x 2  5 x  y  7
Section 8.6: Systems of Inequalities
#1-24: graph each inequality
1) x > 4
2) x > 5
3) 𝑥 ≥ −3
4) 𝑥 ≥ −5
5) x < 2
6) x < 6
7) 𝑥 ≤ 7
8) 𝑥 ≤ −1
9) y > 1
10) y > 3
11) 𝑦 ≥ −3
12) 𝑦 ≥ −6
13) y < -4
14) y < -1
15) 𝑦 ≤ 5
16) 𝑦 ≤ 0
17) x + y < 8
18) x + y < 6
19) 3x + 4y > 12
20) 4x + 3y > 12
21) 𝑥 − 3𝑦 ≤ −6
22) 3x-2y ≤ -6
23) 3𝑥 − 4𝑦 ≥ −24
24) 4𝑥 − 3𝑦 ≥ −24
#25-30: graph each system of inequalities. Label the point of intersection.
25)
3 x  2 y  12
x  y  1
26)
8 x  3 y  26
2 x  y  10
27)
6x  2 y  6
x y 5
28)
4 x  3 y  10
2 x  y  10
29)
5 x  2 y  20
2x  y  9
30)
4 x  2 y  22
3 x  5 y  10
#31-42: graph each system of linear inequalities by hand. Label the corner points.
31)
𝑥+𝑦 ≤9
3𝑥 + 𝑦 ≤ 15
𝑥 ≥ 0, 𝑦 ≥ 0
32)
𝑥+𝑦≤5
3𝑥 + 𝑦 ≤ 9
𝑥 ≥ 0, 𝑦 ≥ 0
33)
𝑥 + 2𝑦 < 8
2𝑥 + 𝑦 ≤ 7
𝑥 ≥ 0, 𝑦 ≥ 0
34)
4𝑥 + 2𝑦 ≤ 12
3𝑥 + 𝑦 ≤ 7
𝑥 ≥ 0, 𝑦 ≥ 0
Section 8.6: Systems of Inequalities
35)
𝑥+𝑦 >4
3𝑥 + 𝑦 > 6
𝑥 ≥ 0, 𝑦 ≥ 0
36)
𝑥+𝑦>5
3𝑥 + 𝑦 > 9
𝑥 ≥ 0, 𝑦 ≥ 0
37)
2𝑥 + 5𝑦 < 20
2𝑥 + 𝑦 ≥ 12
𝑥 ≥ 0, 𝑦 ≥ 0
38)
3𝑥 − 2𝑦 ≤ 5
3𝑥 − 𝑦 ≤ 7
𝑥 ≥ 0, 𝑦 ≥ 0
39)
𝑥 − 2𝑦 < 2
2𝑥 + 𝑦 > 14
𝑥 ≥ 0, 𝑦 ≥ 0
40)
4𝑥 − 2𝑦 ≤ 2
3𝑥 + 𝑦 > 9
𝑥 ≥ 0, 𝑦 ≥ 0
Section 8.7: Linear programming
1)
Maximize: z = 3x + 2y
Subject to:
𝑥+𝑦 ≤9
3𝑥 + 𝑦 ≤ 15
𝑥 ≥ 0, 𝑦 ≥ 0
(Constraints same as problem 31 section 8.6 )
2)
Maximize: z = x + 12y
Subject to:
𝑥+𝑦≤5
3𝑥 + 𝑦 ≤ 9
𝑥 ≥ 0, 𝑦 ≥ 0
(Constraints same as problem 32 section 8.6 )
3)
Maximize: z = 5x + 6y
Subject to:
𝑥 + 2𝑦 < 8
2𝑥 + 𝑦 ≤ 7
𝑥 ≥ 0, 𝑦 ≥ 0
(Constraints same as problem 33 section 8.6 )
4)
Maximize: z = 9x + 20y
Subject to:
4𝑥 + 2𝑦 ≤ 12
3𝑥 + 𝑦 ≤ 7
𝑥 ≥ 0, 𝑦 ≥ 0
(Constraints same as problem 34 section 8.6 )
5)
Minimize: z = 30x + 25y
Subject to:
𝑥+𝑦 ≥4
3𝑥 + 𝑦 ≥ 6
𝑥 ≥ 0, 𝑦 ≥ 0
(Constraints same as problem 35 section 8.6 )
6)
Minimize: z = 10x + 40y
Subject to:
𝑥+𝑦≥5
3𝑥 + 𝑦 ≥ 9
𝑥 ≥ 0, 𝑦 ≥ 0
(Constraints same as problem 36 section 8.6 )
7)
Minimize: z = x + 3y
Subject to:
2𝑥 + 5𝑦 ≤ 20
2𝑥 + 𝑦 ≥ 12
𝑥 ≥ 0, 𝑦 ≥ 0
(Constraints same as problem 37 section 8.6 )
8)
Minimize: z = 50x + 15y
Subject to:
3𝑥 − 2𝑦 ≤ 5
3𝑥 − 𝑦 ≤ 7
𝑥 ≥ 0, 𝑦 ≥ 0
(Constraints same as problem 38 section 8.6 )
9)
Minimize: z = 5x + 4y
Subject to:
𝑥 − 2𝑦 ≤ 2
2𝑥 + 𝑦 ≥ 14
𝑥 ≥ 0, 𝑦 ≥ 0
(Constraints same as problem 39 section 8.6 )
10)
Minimize: z = 3x + 7y
Subject to:
4𝑥 − 2𝑦 ≤ 2
3𝑥 + 𝑦 ≥ 9
𝑥 ≥ 0, 𝑦 ≥ 0
(Constraints same as problem 40 section 8.6 )
Chapter 8 Review:
#1-2: Solve each system of equations using the substitution method
4
1
x y 5
1) 5
4
x  y 1
2)
3x  2 y  5
x  y  41
#3-4: Solve each system of equations using the elimination method.
2 x  3 y  13
3)
5 x  4 y  21
4
2
x  y  12
4) 9
3
x  y  12
#5-6: Solve each system of equations, by hand without matrices
5)
2𝑥 + 4𝑦 − 5𝑧 = −5
−𝑥 + 𝑦 + 2𝑧 = 5
𝑥 − 3𝑦 + 3𝑧 = 4
(pair the middle equation with the other 2 and drop out the x’s)
6)
2𝑥 + 𝑦 − 5𝑧 = 3
−3𝑥 + 2𝑦 + 5𝑧 = 0
𝑥−𝑦+𝑧 =4
(pair the bottom, equation with the other 2 and drop out the y’s)
#7-10: Solve the system of equations using matrices and row operations.
7)
x  2 y  10
2 x  3 y  1
9)
𝑥+𝑦+𝑧 =6
3𝑥 − 2𝑦 + 3𝑧 = 3
5𝑥 − 4𝑦 − 3𝑧 = −5
8)
3 x  2 y  11
x  7 y  10
10)
𝑥−𝑦+𝑧 =3
2𝑥 + 𝑦 + 𝑧 = 13
7𝑥 − 4𝑦 − 5𝑧 = 6
Chapter 8 Review:
#11-12: Solve the system of equations using Cramer’s rule.
11)
4x  2 y  5
2x  5 y  2
12)
𝑥 + 2𝑦 − 𝑧 = 6
3𝑥 + 5𝑦 − 4𝑧 = 13
4𝑥 − 2𝑦 + 𝑧 = 19
#13 -16: Use the following matrices to answer all the problems in this section.
1 0 


A= 3 2


6 1 
3 2 0 
D= 

4  1 3
13) 2A – 3F
1 0  1 


C= 7 2 4


2 3  5
4 5
B= 

1 2 
1  2


F= 5 2


6 7 
2 3 0 


E= 4 1 4


2 6  5
14) E – C
15) AE
16) AB
#17-18: Solve the following systems of equations.
17)
x  7 y  4
x 2  y 2  10
18)
x  2 y  11
x 2  y  13
#19-20: graph each system of linear inequalities by hand. Label the corner points.
19)
𝑥+𝑦 ≤6
2𝑥 + 𝑦 ≤ 11
𝑥 ≥ 0, 𝑦 ≥ 0
20)
𝑥+𝑦≤8
𝑥 + 2𝑦 ≤ 10
𝑥 ≥ 0, 𝑦 ≥ 0