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Math 1324 Test 1 Review

Test instructions

Time: 60 minutes

What is covered? Section 2.1-2.4, 3.1-3.3

Number of questions: 8 Free Response Questions (some questions have sub-questions)

Please show your work neatly. Do not skip steps.

You cannot use any calculator on Test 1.

Remember the make-up policy: NO MAKE UPS!

[10 points] 1. Determine whether the matrix is in row-reduced echelon form.

a. (1 −20 0

|00) b. (

1 0 00 1 00 −9 1

|251)

c. (1 00 10 0

|−5110) d. (

0 0 10 0 01 0 0

|20−4)

e. (1 0 30 1 −10 0 0

|980) f. (

1 0 30 1 5

|−31)

[20 points] 2. The following augmented matrix in row-reduced echelon form is equivalent to the

augmented matrix of a certain system of linear equations. Use this result to solve the system of

equations. Write the answer in the ordered pairs.

a. (1 −10 30 0

|

31

−1

3

)

b. (1 00 10 0

|230)

c. (1 0 10 1 −1

|−3−17

)

d. (1 0 00 1 00 0 1

|1−87)

[10 points] 3. Solve the system of linear equations using the Gauss-Jordan elimination method.

a. {

𝑥 + 3𝑦 + 𝑧 = 3𝑦 + 2𝑧 = 4

−9𝑦 + 2𝑧 = −16

b. {

𝑥 − 2𝑦 = 27𝑥 − 14𝑦 = 143𝑥 − 6𝑦 = 6

[5 points] 4. Solve for z.

(𝑥 −20 3464 𝑧 −2710 8 24

) + 3(3 −5 −3−8 10 4−3 𝑦 7

) = −5(9 7 −5−8 8 3𝑤 1 −9

)

[20 points] 5. Compute the following if possible.

a. −4(0 2−3 32 1

) + (−6 0−2 −110 12

)

b. (−2 1 30 5 6

) + (1 37 −1−4 0

)

c. (−1 2 −35 0 1

)(1 4−6 2−3 10

)

d. (−4 −1 50 −2 3

) (2 0 61 −3 1

)

[10 points] 6. Graph the feasible region for the following system of inequalities.

{

𝑥 ≤ 4𝑦 ≤ 9

𝑥 + 𝑦 ≥ 9𝑥 ≥ 0𝑦 ≥ 0

[15 points] 7. Solve the following linear programming problem:

Maximize 𝑃 = 4𝑥 + 6𝑦 subject to {

2𝑥 + 5𝑦 ≤ 20𝑥 + 𝑦 ≤ 7𝑥 ≥ 0𝑦 ≥ 0

[10 points] 8. The officers of a high school senior class are planning to rent buses and vans for

a class trip. Each bus can transport 40 students, requires 3 chaperones, and costs $1,200 to

rent. Each van can transport 8 students, requires 1 chaperone, and cost $100 to rent. The

officers must plan to accommodate at least 400 students. Since only 36 parents have

volunteered to serve as chaperones, the officers must plan to use at most 36 chaperones. How

many vehicles of each type should the officers rent in order to minimize the transportation

costs? What are the minimal transportation costs?

a. Define your variables.

b. Construct and fill in a table.

Transportation Information Buses Vans

Number of vehicles

Costs

c. State the Linear Programming Problem. Do Not Solve.