Riva – Adv Precalculus
Chapter 7 Review
1) Use Gaussian elimination (i.e., reduce the augmented coefficient matrix to
row-echelon form) to solve the following systems of equations. (Note: Make
sure you are able to do these problems by hand!)
a.
d.
6y + 4z = -18
x - 2y - z + 2w = 6
3x + 3y = 9
2x + 2y - 2z = 10
b.
2x - 3z = 12
2x + y - 3z = 4
4x + 2z = 10
-2x + 3y -13z = -8
c.
10x - 3y + 2z = 0
19x - 5y - z = 0
x + y - w = -1
x + y - 2z + w = 1
e.
3x + 3y +12z = 6
x + y + 4z = 2
2x + 5y + 20z = 10
-x + 2y + 8z = 4
2) Describe the row-echelon form of an augmented matrix that corresponds to a
system of linear equations that is inconsistent.
3) Find the equation of the circle 𝑥 2 + 𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 that passes
through the points (-6,-1), (-4,3), and (2,-5).
4) The Augusta National Golf Club in Augusta, GA is an 18-hole course that
consists of par-3 holes, par-4 holes, and par-5 holes. A golfer who shoots par
has a total of 72 strokes for the entire course. There are two more par-4
holes than twice the number of par-5 holes, and the number of par-3 holes is
equal to the number of par-5 holes. Find the number of par-3, par-4, and par5 holes on the course.
a. Write a system of linear equations that represents the situation
b. Write a matrix equation that corresponds to your system.
c. Solve your system of linear equations using Gaussian elimination (i.e.,
reducing the augmented coefficient matrix to row-echelon form).
1 3
5) Let 𝐴 = [
]
2 4
2
a. Find 𝐴
b. Find 𝐴−1
1
2
−1
6) Let 𝐴 = [ 3
7 −10]. Find 𝐴−1 by hand.
−5 −7 −15
Riva – Adv Precalculus
7) Water flowing through a network of pipes (in thousands of cubic meters per
hour) is shown below.
a. Use matrices to solve this system for the water flow represented by
𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 .
b. Find the network flow pattern when 𝑥6 = 0 and 𝑥7 = 0.
c. Find the network flow pattern when 𝑥5 = 1000 and 𝑥6 = 0.
é 0 2 ù
é 2 -1 ù
é 1 1 1 ù
é 3 -2 3 ù
ê
ú
ê
ú
8) Let A = ê -1 8 ú , B = ê 1 3 ú, C = ê
ú, D = ê
ú.
2 3 4 û
1 -2 1 û
ë
ë
êë 6 4 úû
êë 1 4 úû
Find (if possible): (Make sure you are able to perform these calculations by
hand)
a. A – B
b. -4C
c. D+2A
d. DA
e. AD
f. AB
g. 𝐴−1
9) Write the inverse of the coefficient matrix of the following system of
equations. Then use it to solve the system of equations. (You can use your
calculator on this problem)
2x + 5y + w = 11
x + 4y + 2z - 2w = -7
2x - 2y + 5z + w = 3
x - 3w = -1
Riva – Adv Precalculus
10)The number of calories burned by individuals of different weights
performing different types of exercises for 20-minute periods are shown in
the matrix.
a. A 120-pound person and a 150-pound person bicycle for 40 minutes,
jog for 10 minutes, and walk for 60 minutes. Organize a matrix A for
the time spent exercising in units of 20-minute intervals.
b. Find the product AB
c. Explain the meaning of the product AB in the context of the situation.
11) A coffee manufacturer sells a 15-pound package of coffee for $44 that
contains five flavors of coffee. French vanilla coffee costs $2 per pound,
hazelnut coffee costs $2.50 per pound, Swiss chocolate flavored coffee costs
$3 per pound, caramel flavored costs $3.50 per pound, and Blue Mountain
coffee costs $4 per pound. The package contains the same amount of
hazelnut as Swiss chocolate. The package contains twice as much French
vanilla as Blue Mountain. The package has three more pounds of caramel
than Blue Mountain. Let f represent the number of pounds of French vanilla,
h represent the number of pounds of hazelnut, s represent the number of
pounds of Swiss chocolate, c represent the number of pounds of caramel, and
b represent the number of pounds of Blue Mountain.
a. Write a system of linear equations that represents the situation
b. Write a matrix equation that corresponds to your system.
c. Solve your system of linear equations using an inverse matrix. Find
the number of pounds of each flavor of coffee in the 15-pound
package.