Download Chapter-8-problems

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, 𝑥 𝑖𝑠 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟}
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, 𝑥 𝑖𝑠 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟}
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 just say it has infinitely many solutions.
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
50)
𝑥 − 2𝑦 − 𝑧 = 8
2𝑥 − 3𝑦 + 𝑧 = 23
4𝑥 − 5𝑦 + 5𝑧 = 53
51)
𝑥 + 2𝑦 − 𝑧 = 8
−𝑥 − 3𝑦 + 𝑧 = 23
𝑥 + 𝑦 + 5𝑧 = 39
(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)
(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 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
20)
𝑥−𝑦+𝑧 =2
2𝑥 + 𝑦 + 𝑧 = 5
5𝑥 − 4𝑦 + 𝑧 = 6
7𝑥 + 4𝑦 − 𝑧 = 9
Section 8.3: Systems of Linear Equations – Determinants
#1 – 8: Find the value of the determinant of the following matrices
  2 4
1
3
4 5

1 2 
1) 
3 5  1


5) 4 2 3 6)


2 5 6 
3  5 

0  2 
2) 
1 0  1 
7 2 4 


2 3  5
3) 
 3 2 8 


7) 0 6  3


2 1 4 
 4 5

 1 0 
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)
13)
15)
17)
19)
4x  y  2
2)
x3  2  y  0
 2 x  y  5
4)
x 2  y 2  25
x2  y  0
6)
8x 2  4 x  y  0
y  2 x 2  2
8)
y  2x 4  4x 2  2
3x  y  4
x2  y  0
x2  y  0
x y 0
y  x
y  x 3  4x 2  2x
x  7 y  6  12
x 2  y 2  36
x  7 y  6  4
x 2  y 2  20
3x  y  8
x2  2  y  6
10)
12)
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
y  x
14)
y  x 3  5 x 2  3x
16)
18)
20)
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)
27)
6x  2 y  6
x y 5
28)
29)
5 x  2 y  20
2x  y  9
30)
4x  y  4
2 x  y  12
4 x  2 y  10
2x  y  8
4 x  2 y  22
3 x  5 y  10
Section 8.6: Systems of Inequalities
#31-40: 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
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.
4
2
x  y  12
4) 9
3
x  y  12
2 x  3 y  13
3)
5 x  4 y  21
#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
Problems 9 and 10 have been deleted.
8)
3 x  2 y  11
x  7 y  10
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) Solve the following systems of equations.
x  2y  5
x 2  y  10
18) Graph the system of linear inequalities by hand. Label the corner points.
𝑥+𝑦 ≤6
2𝑥 + 𝑦 ≤ 10
𝑥 ≥ 0, 𝑦 ≥ 0
19) There is no problem 19
20) 𝑥 + 𝑦 ≤ 8
𝑥 + 2𝑦 ≤ 10
𝑥 ≥ 0, 𝑦 ≥ 0
Grima MAT 151
Chapter 8 Practice Test
#1-2: Solve each system of equations using either the substitution method or the elimination method,
0 points if no work is shown even if answer is correct.
2
1
x y 3
1) 3
4
x  y 1
2)
3 x  2 y  13
x  5 y  7
3) Solve each system of equations, by hand without matrices, 0 points if no work is shown even if
answer is correct.
2𝑥 + 4𝑦 − 5𝑧 = 17
−𝑥 + 𝑦 + 2𝑧 = −5
𝑥 − 3𝑦 + 3𝑧 = −2
(pair the middle equation with the other 2 and drop out the x’s)
4) Solve the system of equations using matrices and row operations. 0 points if no matrix work is
shown even if answer is correct.
3x  2 y  16
2 x  3 y  11
#5-6 Use the following matrices to answer all the problems in this section. You may use a calculator to
solve these two problems.
4 5

1 2 
1 0 


A= 3 2


6 1 
3 2 0 
D= 

4  1 3
5) 2D – C
7) has been deleted
B= 
6) AC
1 0  1

7 2 4 
C= 
8) Solve the system of equations using Cramer’s rule, 0 points if solved with another method, even if
answer is correct.
3x  2 y  4
2x  3y  7
D = _________________________________
Dx = _________________________________
Dy = _________________________________
X =_________ y = ______________
#9-10: Solve the following systems of equations.
9)
x y 4
x y 5
10) x 2  y  10
x 2  y 2  13
11) graph the system of linear inequalities by hand. Label the corner points.
𝑥+𝑦 ≤6
2𝑥 + 𝑦 ≤ 10
𝑥 ≥ 0, 𝑦 ≥ 0
Answers: 1) (3,4)
6)
1 0
[17 4
13 0
9)
(2,3) (3,2)
−1
5]
2
2) (3,2)
3) (4,1,-1)
4) (2,5)
7) deleted 8) D = 13
10) (-2,6) (3,1)
5 4 1
1  4
5) 
Dx = 26

2
Dy = 13 x = 2 y = 1
Grima MAT 151 Chapter 8 in class practice problems
#1-2: Solve each system of equations using either the substitution method or the elimination method,0
points if no work is shown even if answer is correct.
1)
5 x  3 y  29
x  y 1
2)
3x  2 y  6
x  5 y  19
3) Solve each system of equations, by hand without matrices, 0 points if no work is shown even if
answer is correct.
𝑥 + 4𝑦 − 5𝑧 = 6
−𝑥 + 𝑦 + 2𝑧 = 1
𝑥 − 2𝑦 + 3𝑧 = 2
(pair the middle equation with the other 2 and drop out the x’s)
4) Solve the system of equations using matrices and row operations. 0 points if no matrix work is
shown even if answer is correct. (I hope this hint helps more than it hurts. To solve this problem
correctly, you should write the system as a matrix. On step one you should eliminate an x, then in step 2
eliminate a y.)
x  2y  7
 x  y  1
#5-6 Use the following matrices to answer all the problems in this section. You may use a calculator to
solve these two problems.
1 0 

3 2 
4 5

1 2 
A= 
5) 3A + 2B
B= 
6) AC
1 0  1

7 2 4 
C= 
7) Solve the system of equations using Cramer’s rule, 0 points if solved with another method, even if
answer is correct.
3x  2 y  5
5 x  3 y  21
D = _________________________________
Dx = _________________________________
Dy = _________________________________
X =_________ y = ______________
#8-9: Solve the following systems of equations.
8)
y  x  1
x 2  2 y  22
9)
2x  y  5
x 2  y  4 (hint solve the top equation for y the substitute into the bottom equation)
10) graph the system of linear inequalities by hand. Label the corner points.
𝑥+𝑦 ≤6
𝑥 + 2𝑦 ≤ 8
𝑥 ≥ 0, 𝑦 ≥ 0
Answers:
1) (4,3)
11 10 

11 10 
5) 
2) (4,3)
3) (3,2,1)
1 0  1

17 4 5
6) 
7) D = 19 Dx = 57 Dy = 38 x = 3 y = 2
8) (-6, -7) (4, 3)
9) (1,3)
10)
4) (3,2)