# System of linear equations and augmented matrices

MTH212

Unit 3 - Individual Project A

1. To raise money, the local baseball teams decided to sell team logo hats (H) and T-shirts (T). The league director decided to hold a contest among the teams to see which team can raise the most money. The contest lasted for 3 weeks. Here are the results of the first 2 weeks. The numbers represent the number of hats and T-shirts sold.

A = [ ] B = [ ]

A. How much of each item had the teams sold by the end of the second week. Use matrices to solve the problem. Final answer must be given in matrix form. Show all work to receive full credit.

B. Which team had sold the most items at the end of the second week, and how many total items did they sell?

C. By the end of the third week, the totals were as follows:

D. Which baseball team won the contest, and what was their total sales?

2. Use augmented matrices to solve the following systems of equations. Show all work to receive full credit. Final answer must be given in matrix form.

A. 3x + 4y = 11

x + 3y = 2

B. x - 3y =5

4x -12y=13

3. A company's employees are working to create a new energy bar. They would like the two key ingredients to be peanut butter and oats, and they want to make sure they have enough carbohydrates and protein in the bar to supply the athlete. They want a total of 22 carbohydrates and 14 grams of protein to make the bar sufficient. Using the following table, create a system of two equations and two unknowns to find how many tablespoons of each ingredient the bar will need. Solve the system of equations using matrices. Show all work to receive full credit.

Carbohydrates Protein

Peanut Butter 2 4

Oats 8 1

A. Write an equation for the total amount of carbohydrates.

B. Write an equation for the total amount of protein.

C. Determine the augmented matrix that represents the previous two equations.

D. Solve for the previous matrix. Show all work to receive full credit.

E. How many tablespoons of each will there need to be for the new energy bar?

4. A total of 700 tickets were sold for a musical. Senior citizen tickets sold for $15, children tickets sold for $20, and adult tickets sold for $25; the total earnings from ticket sales was $15,750. Five times more children tickets were sold than senior citizen tickets. How many tickets of each type were sold? Set up a system of three equations and three unknowns, use an augmented matrix to solve, and show all work to receive full credit.

A. What are the three unknowns?

B. Write a separate equation representing each of the first three sentences of the word problem.

C. Determine the augmented matrix that represents the three equations.

D. Solve for the matrix. Show all work to receive full credit.

E. How many of each type of ticket were sold?

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#### Solution Preview

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1. To raise money, the local baseball teams decided to sell team logo hats (H) and T-shirts (T). The league director decided to hold a contest among the teams to see which team can raise the most money. The contest lasted for 3 weeks. Here are the results of the first 2 weeks. The numbers represent the number of hats and T-shirts sold.

A. How much of each item had the teams sold by the end of the second week. Use matrices to solve the problem. Final answer must be given in matrix form. Show all work to receive full credit.

A + B =

At the end of the second week,

Bears sold 25 hats and 55 T-shirts

Charges sold 17 hats and 52 T-shirts

Tigers sold 22 hats and 89 T-shirts

Blue jays sold 25 hats and 55 T-shirts

B. Which team had sold the ...

#### Solution Summary

This solution file contains detailed and well-written answers to four different questions on finding the solution to a system of linear equations. First question involves the matrix operations such as matrix addition and subtraction. Second question involves the solution to a system of linear equations using augmented matrix. Third and forth questions were word problems. To solve these, I have constructed a system of linear equations using the given information, and then solved them using their augmented matrices. Answers were written with the aid of equation editor making them look neat and easy to read. This is an excellent solution set to learn the techniques of solving system of equations using augmented matrices.