Week 3: September 27, 2023
Exercises for Test 3 on Gauss-Jordan elimination
1. (2.1 #5.) Use the echelon method to solve the following system of two equations in two unknowns.
x+ y=5
2x − 2y = 2
2. (2.1 #6.) Use the echelon method to solve the following system of two equations in two unknowns.
4x + y = 9
3x − y = 5
3. (2.1 #31.) Solve the following system of equations.
x + 2y + 3z = 11
2x − y + z = 2
4. (2.1 #39.) If 20 lb of rice and 10 lb of potatoes cost $16.20, and 30 lb of rice and 12 lb of
potatoes cost $23.04, how much will 10 lb of rice and 50 lb of potatoes cost?
5. (2.1 #42.) An apparel shop sells pants for $45 a pair and shirts for $35 each. Its entire stock is
worth $51, 750. But sales are slow and only half the pants and two-thirds of the shirts are sold,
for a total of $30, 600. How many pairs of pants and how many shirts are left in the store?
6. (2.2 #5.) Write the augmented matrix for the following system of linear equations. Do not
solve.
3x + y = 6
2x + 5y = 15
7. (2.2 #7.) Write the augmented matrix for the following system of linear equations. Do not
solve.
2x + y + z= 3
3x − 4y + 2z = −7
x+ y + z= 2
8. (2.2 #11.) Write the system of equations associated with the following augmented matrix if the
variables are x, y and z.
1 0 0 4
0 1 0 −5
0 0 1 1
9. (2.2 #21.) Use the Gauss-Jordan method to solve the following system of equations.
x+ y= 5
3x + 2y = 12
10. (2.2 #22.) Use the Gauss-Jordan method to solve the following system of equations.
x + 2y = 5
2x + y = −2
11. (2.2 #30.) Use the Gauss-Jordan method to solve the following system of equations.
x = 1−y
2x = z
2z = −2 − y
1
12. (2.2 #52.) The U-Drive Rent-A-Truck Company plans to spend $7 million on 200 new vehicles.
Each van will cost $35, 000, each small truck $30, 000, and each large truck $50, 000. Past
experience shows that they need twice as many vans as small trucks. How many of each kind of
vehicle can they buy?
13. (2.2 #68.) At rush hours, substantial traffic congestion is encountered at the traffic intersections
shown in the figure. (The streets are one-way, as shown by the arrows.)
300 900
x1
M Street
700 A B 200
x4 x2
N Street x3
200 D C 400
400 300
10th Street 11th Street
The city wishes to improve the signals at these corners so as to speed the flow of traffic. The
traffic engineers first gather data. As the figure shows, 700 cars per hour come down M Street
to intersection A, and 300 cars per hour come down 10th Street to intersection A. A total of
x1 of these cars leave A on M Street, and x4 cars leave A on 10th Street. The number of cars
entering A must equal the number leaving, so that
x1 + x4 = 700 + 300
or
x1 + x4 = 1000.
For intersection B, x1 cars enter on M Street and x2 on 11th Street. The figure shows that 900
cars leave B on 11th and 200 on M. Thus,
x1 + x2 = 900 + 200
x1 + x2 = 1100.
(a) Write two equations representing the traffic entering and leaving intersections C and D.
(b) Use the four equations to set up an augmented matrix, and solve the system by the Gauss-
Jordan method.
(c) Based on your solution to part (b), what are the largest and smallest possible values for the
number of cars leaving intersection A on 10th Street?
(d) Answer the question in part (c) for the other three variables.
(e) Verify that you could have discarded any one of the four original equations without changing
the solution. What does this tell you about the original problem?
2
14. (2.5 #21.) Find the inverse, if it exists, for the matrix.
1 0 0
0 −1 0
1 0 1
15. (2.5 #23.) Find the inverse, if it exists, for the matrix.
−1 −1 −1
4 5 0
0 1 −3
16. (2.5 #31.) Solve the following system of equations by using the inverse of the coefficient matrix.
2x + 5y = 15
x + 4y = 9
17. (2.5 #39.) Solve the following system of equations by using the inverse of the coefficient matrix.
−x − y − z = 1
4x + 5y = −2
y − 3z = 3
