Pre-Calculus
Chapter 7 Review
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Date ___________________
Hour ______
7.1 SUBSTITUTION
Solve the system by the method of substitution. Show all work.
𝑥+𝑦 =2
𝑥2 − 𝑦2 = 9
1. {
{
2.
𝑥−𝑦 =0
𝑥−𝑦 =1
3. {
𝑦 = 2𝑥 2
𝑦 = 𝑥 4 − 2𝑥 2
4. Geometry The perimeter of a rectangle is 480 meters and its length is 1.5 times its width. Find the
dimensions of the rectangle.
7.2 ELIMINATION
Solve the systems using elimination. Show all work.
1
3
7
2𝑥 − 𝑦 = 2
3𝑥 − 2𝑦 = 0
𝑥+ 𝑦=
5. {
7. {
5
10
50
6𝑥 + 8𝑦 = 39
3𝑥 + 2(𝑦 + 5) = 10
6. { 2
1
1
𝑥+ 𝑦=
5
2
8. {
1.25𝑥 − 2𝑦 = 3.5
5𝑥 − 8𝑦 = 14
5
9. Airplane Speed Two planes leave Pittsburgh and Philadelphia at the same time, each going to the
other city. One plane flies 25 miles per hour faster than the other. Find the airspeed of each plane if the
cities are 275 miles apart and the planes pass each other after 40 minutes of flying time.
7.3 & 7.4 MULTI-VARIABLE SYSTEMS
Use back substitution to solve the system of equations.
3𝑥 − 𝑦 + 5𝑧 = −10
10. { 4𝑦 − 2𝑧 = 16
−3𝑧 = 18
Determine the order of the following matrices.
−9 0
2 6
7 8 6
]
11. [
12. [
5 2
1 0 4
7 0
−8
]
0
Write the augmented matrix for the system.
5𝑥 + 2𝑦 + 3𝑧 = 2
13. { 5𝑥 + 𝑦
=4
8𝑥
+ 10𝑧 = −1
Solve each system using Row Echelon Form or Matrices. Include all notations/necessary steps.
4𝑥 + 2𝑦 + 3𝑧 = 11
𝑥 − 3𝑦 + 2𝑧 = 5
15. { 𝑥 − 2𝑦 + 𝑧 = 6
14. { 𝑥 + 2𝑦 + 𝑧 = 7
2𝑥 + 𝑦 + 2𝑧 = 7
2𝑥 − 𝑦 + 5𝑧 = 18
7.5 MATRICES
Find x and y.
𝑥+3
4
0
−3
16. [
−2
𝑦+5
−4𝑦
5𝑥 − 1
2 ]=[ 0
6𝑥
−2
4
−3
16
−44
2 ]
6
Find, if possible, (a) A + B, (b) A – B, (c) 4A, and (d) A + 3B
7 3
10 −20
6
], 𝐵 = [
]
17. 𝐴 = [
18. 𝐴 = [5
−1 5
14 −3
3
0
−1
2
7
0
2], 𝐵 = [−4
3
2
5
8
−1
1
6]
1
19. Sales At a dairy mart, the numbers of gallons of skim, 2%, and whole milk sold on Friday,
Saturday, and Sunday of a particular week are given b the following matrix.
Skim 2% Whole
milk milk milk
40
𝐴 = [60
76
64
82
96
52 𝐹𝑟𝑖𝑑𝑎𝑦
76] 𝑆𝑎𝑡𝑢𝑟𝑑𝑎𝑦
84 𝑆𝑢𝑛𝑑𝑎𝑦
A second matrix gives the selling price per gallon and the profit per gallon for each of the three types
of milk sold by the dairy mart.
Selling
price per Profit
gallon per gallon
2.65
𝐵 = [2.81
2.93
0.25 𝑆𝑘𝑖𝑚 𝑚𝑖𝑙𝑘
0.30] 2% 𝑚𝑖𝑙𝑘
0.35 𝑊ℎ𝑜𝑙𝑒 𝑚𝑖𝑙𝑘
(a) Find AB. What is the meaning of AB in the context of the situation?
(b) Find the dairy mart’s profit for Friday through Sunday.
7.6 INVERSE MATRIX
Use an inverse matrix to solve (if possible) the system of linear equations.
𝑥 + 5𝑦 = −1
18. {
3𝑥 − 5𝑦 = 5
3𝑥 + 2𝑦 − 𝑧 = 6
𝑥
19. { − 𝑦 + 2𝑧 = −1
5𝑥 + 𝑦 + 𝑧 = 7
𝑥 + 2𝑦 + 𝑧 − 𝑤 = −2
2𝑥 + 𝑦 + 𝑧 + 𝑤 = 1
20. {
𝑥 − 𝑦 − 3𝑧
=0
𝑧+𝑤 =1
21. Fast-Food Sales A small fast-food chain with restaurants in Santa Monica, Long Beach, and
Anaheim sells only hamburgers, hot dogs, and milk shakes. On a certain day, sales ere distributed
according to the following matrix.