# Linear Programming, Gaussian Elimination and Matrix

Please see the attached files for the fully formatted problems.

8. (6 pts) There is a jar containing nickels and quarters.

Suppose there are 89 coins in the jar, and the total value is $19.45.

We would like to find out how many nickels and how many quarters are in the jar.

Let x = number of nickels and y = number of quarters.

Write a system of two linear equations involving variables x and y, corresponding to the information given about the coins. [Just set up the system of linear equations. You are not asked to solve the system.]

9. (20 pts)

Consider the system of three linear equations:

Template:

(If you wish, you can use the Equation Editor template shown above, to make the typing easier. Just copy, paste where appropriate, and edit with the Equation Editor as needed.)

(a) Write the augmented 3 x 4 matrix corresponding to this system.

(b) Solve the system by using the Gaussian elimination method. Show work. (When you are done, it is a very good idea to check your solution by substitution into the original system of equations, but do not go to the trouble of submitting your check.)

10. (6 pts) Let U = {1, 2, 3, 4, 5, 6, 7}, A = {1, 3, 5, 6} and B = {1, 2, 6}.

List the elements of the indicated sets.

(a)

(b)

(Be sure to notice the complement symbol applied to A)

11. (8 pts) In a certain state legislature of 120 legislators, there are 60 Democrats, 40 Republicans, and 20 Independents.

The following matrix summarizes (in decimal format) the percentage of legislators who voted for or against a particular finance bill, by affiliation.

For instance, 70% of Republican legislators voted against the finance bill.

(a) Determine a matrix calculation resulting in a matrix having two entries, where the first entry is the vote total for the bill and the second entry is the vote total against the bill.

State the matrix calculation and the state the entries in the resulting matrix. (For example, a matrix calculation for matrices A and B might be A + B or A - B or AB or BA, etc. What is the matrix calculation involving L and P in this problem? Show the steps in the calculation and the result.)

(b) A bill passes if a majority (over half) of the legislators vote for the bill. By looking at your results from (a), did the bill pass or not?

12. (25 pts) Two kinds of cargo, A and B, are to be shipped by truck. Each crate of cargo A is 50 cubic feet in volume and weighs 200 pounds, whereas each crate of cargo B is 10 cubic feet in volume and weighs 360 pounds. The shipping company earns $80 per crate for cargo A and $100 per crate for cargo B. The truck has a maximum load limit of 1,000 cubic feet and 7,200 pounds. The shipping company would like to earn the highest revenue possible.

(a) Fill in the chart below as appropriate.

Cargo A

(per crate)

Cargo B

(per crate)

Truck Load Limit

Volume

Weight

Revenue

(b) Let x be the number of crates of cargo A and y the number of crates of cargo B shipped by one truck. Using the chart in (a), give two inequalities that x and y must satisfy because of the truck's load limits.

(c) Give two inequalities that x and y must satisfy because they cannot be negative.

(d) Give an expression for the total revenue earned from shipping x crates of cargo A and y crates of cargo B.

(e) State the linear programming problem which corresponds to the situation described. That is, be sure to indicate whether you have a maximization problem or a minimization problem, and state the objective function and the constraints. (This part is mostly a summary of the previous parts)

(f) List the inequalities from part (e) in standard form.

(g) Solve the linear programming problem. You will need to find the feasible set and determine the vertices. You do not have to submit your graph, and you do not have to show your algebraic work in finding the vertices, but you must list your vertices and corresponding values of the objective function.

Vertex (x, y) Objective Function

(h) Write your conclusion with regard to the word problem. State how many crates of cargo A and how many crates of cargo B should be shipped in one truck, in order to earn the highest total revenue possible. State the value of that maximum revenue.

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#### Solution Summary

This solution contains step by step answers to linear programming, Gaussian Elimination, Systems of Equations, Inequlaities, Matrix and Set Operations.

Matrices, Inverse, Transpose, Determinant, Gaussian Elimination and Cramer's Rule

Matrix methods can be used to solve linear programming problems. A linear programming problem is used to find an optimal solution, subject to stated restraints.

1. For example, consider an accountant who prepares tax returns. Suppose a form 1040EZ requires $12 in computer resources to process and 22 minutes of the accountant's time. Assume a form 1040A takes $25 in computer resources and needs 48 minutes of the accountant's time. If the accountant can spend $630 on computer resources and has 1194 minutes available, how many forms of 1040EZ and 1040A can the accountant process?

2. You are given the following system of linear equations:

x - y + 2z = 13

2x + 2y - z = -6

-x + 3y + z = -7

a. Provide a coefficient matrix corresponding to the system of linear equations.

b. What is the inverse of this matrix?

c. What is the transpose of this matrix?

d. Find the determinant for this matrix.

3. Calculate the following for

a. A * B 2 -3 -4 1

b. -4A A= and B=

c. AT 7 4 6 5

4. Solve the following linear system using Gaussian elimination.

Show work.

3x + y - z = -5

-4x + y = 6

6x - 2y + 3z = 2

5. Solve the following linear system for x using Cramer's rule.

Show work.

x + 2y - 3z = -21

2x - 6y + 8z = 73

-x - 2y + 4z = 28