# Linear Equations, Linear Programming and Matrix Inverse

Hello, I would like for someone to check my answers for the problems that are highlighted in green to make sure that I have solved them correctly. However, there are several problems that are highlighted in yellow that I need help with. Could someone please provide the formula and break-down each step for those problems? Also, please use word instead of excel? Thanks.

1. Find the slope.

The line that passes through points (2, 1) and (4, 6).

(y2 - y1)/(x2 - x1) = (-6 - 1)/(4 - 1) = -7/3

2. Find the slope.

2x + y = 4

Y = -2x + 4

Slope is m = -2

3. Find a general equation for the line having the given properties.

Slope = 4; containing (3, 4)

M = 4

(x1 - y1) = (-3 - 4)

Y - y1 = m(x - x1)

Y - 4 = 4(x + 3)

Y - 4 = 4x + 12

Y = 4x + 16

4x - y + 16 = 0

4. Without solving, determine whether the lines below are perpendicular or parallel.

The slopes are parallel because they are equal to each other with slopes of 2/3.

5. A manufacturer produces items at a daily cost of $1.25 per item and sells them for $2 per item. The daily operational overhead is $450. What is the break-even point? Remember a point has an x and a y value.

Let x items correspond to the break-even level.

Income = Cost

2x = 1.25x + 450

2x - 1.25x = 450

0.75x = 450

X = 600 items for break-even

2(600) = 1200

(600, 1200) break-even point coordinates

6.

Write the augmented matrix of

3x + y = 4 x - 3y = 8

Do not solve.

3 1 │ 4

1 3 │ 8

7. Perform the row operation R2 = (2)r1 + r2 on the matrix

Second row is changed

= 6 - 2(1) = 4

= -1 - 2(5) = -11

= 2 - 2(-6) = 14

New Matrix

1 5 -6

4 -11 14

8. Write the solution of the following matrix.

X = -1, y = 0, and z = 3

9. What is the dimension of the matrix shown below?

The dimension of this matrix is 4 x 3 because it has 4 rows and 3 columns.

10. Find:

4 -12 4 8 -1 1 -4 -11 3

- =

0 8 4 2 0 1 -2 8 3

11. a. If A is a matrix of dimension 2 × 3 and B is a matrix of dimension 3 × 5, is AB defined?

Yes, AB would be defined because A has 3 columns and B has 3 rows.

b. If so, what is its dimension?

The dimensions of AB will be 2 x 5

c. Is BA defined?

BA is not defined because the number of columns in B doesn't equal the number of rows in A.

d. If so, what is its dimension?

There are no dimensions for BA because it's not defined.

12. Find:

13. Find the inverse of:

14. Compute the transpose of A =

A = AT = 2 1

3 -1

0 5

15. Introduce slack variables and set up the initial tableau. Do not solve.

Maximize

subject to the constraints

X1 + 3x2 + 2x3 + S1 = 10

4x1 + 2x2 + 3x3 + 0 + S2 = 8

-8x1 - 2x2 - 3x2 + Z = 0

16. Find the Pivot element for

Do not Solve.

The pivot element would be ½ located in columns 2 (x2), row 1.

17. Find the solution of the following final tableau

18. Write the dual problem for the following. DO NOT SOLVE.

Minimize

subject to the constraints

Maximize D = 2y1 + 6y2

Subject to the constraints: y1 + 3y2 ≤ 3

Y1 + 2y2 ≤ 5

Y1 ≥ 0, y2 ≥ 0

19. A company makes three products, A, B, and C. There are 500 pounds of raw material available. Each unit of product A requires 2 pounds of raw material, each unit of product B requires 2 pounds of raw material, and each unit of product C requires 3 pounds of raw material. The assembly line has 1,000 hours of operation available. Each unit of product A requires 4 hours of assembly, while each unit of products B and C requires 5 hours. The company realizes a profit of $500 for each unit of product A, $600 for each unit of product B, and $1,000 for each unit of product C. How many units of each of the products should the company make to maximize the profit? DO NOT SOLVE THIS PROBLEM. JUST SET UP THE PROBLEM AND WRITE THE INITIAL TABLEAU WITH THE SLACK VARIABLES.

Use the following to answer questions 20-22:

A $152,400 loan is taken out at 11.5% for 25 years, for the purchase of a house. The loan requires monthly payments.

PV = 152,400, Interest = .115/12 = .0096, N = 25 x 30 = 300

20. Find the amount of each payment.

PMT = PV i(1 + i)n .

(1 +i)n - 1

= 152,000 .0096(1 + .0096)300

(1 + .0096)300 - 1

= 152,000 .0096(17.57125)

16.57125

= 152,000 .168684

16.57125

= 152,000 * 0.010179

= $1,547.21

21. Determine the total amount repaid over the life of the loan.

$1,547.21 * 300 = $464,163

22. Find the total interest paid over the life of the loan.

$464,163 - 152,000 = $312,163

23. In the game Over-and-Under, a pair of dice is rolled and one bets $1 whether the sum of dots showing on the two dice is over 7, under 7, or exactly 7 with payoffs of $2, $2, and $5, respectively. Determine a person's expected net winnings if the bet is as indicated.

a. Over 7

b.

Under 7

c. Exactly 7

d. Is the game fair?

24. A box contains 4 defective and 8 good light bulbs. If 3 bulbs are selected at random. What is the probability that exactly one is good?

25. Four percent of the items coming off an assembly line are defective. If the defective items occur randomly and ten items are chosen for inspection, what is the probability that exactly two items are defective?

26. The following data represent the number of car accidents per month in a small town over a two-year period.

Number of Accidents

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec -1

Yr 1 169 163 170 165 165 169 168 172 170 172 171 165 -1

Yr 2 170 168 177 164 173 166 173 176 177 172 170 172 -1

Construct a frequency distribution based on the intervals 163 - 165, 166 - 168, 169 - 171, 172 - 174, 175 - 177.

Use the following to answer questions 27-31:

Use the following sample: 28, 30, 24, 30, 32, 40, 22, 25, 26, 34

27. Find the mean.

28 + 30 + 24 + 30 + 32 + 40 + 22 + 25 + 26 + 34 = 291/10 = 29.1 or 29

10

28. Find the median.

22, 24, 25, 26, 28, 30, 30, 32, 34, 40

28 + 30 = 58 = 29

2 2

29. Find the mode.

The mode is 30 because it occurs more often than any other number.

30. Find the standard deviation for a sample.

22, 24, 25, 26, 28, 30, 30, 32, 34, 40

22 484

24 576

25 625

26 676

28 784

30 900

30 900

32 1,024

34 1,156

40 1,600

Total = 8,725

S = 8,725 - 10(29)2

10 - 1

S = 315/9 = 35

= 5.92

31. Find the z-score for 30. (Assume a normal distribution.)

32. A probability distribution has an expected value of 28 and a standard deviation of 4. Find the area under the standard normal curve between 20 and 36

Use the following to answer questions 33-34:

Using z-scores, what is the area under the standard normal curve?

33. To the left of 1.8?

P(z ≤ 1.8) = .9641

34. Between 2.20 and 1.36

P(-220 ≤ z ≤ 1.36) = .9131 - .0139 = .8992

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#### Solution Summary

Step by step solutions to all the problems are provided.

Matrices - Matrix methods can be used to solve linear programming problems. For specific problems, please see the posted problems.

3-6 pages

1. 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.

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.

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 = -22

2x - 6y + 8z = 74

-x - 2y + 4z = 29

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