Practice Open-Ended Question (10 mins)

“Keeping Records at the Gas Station”

Juan has a part-time job at a gas station. Part of his responsibility is to keep track of the sales of stock items. During one week last July he sold the following:

160 quarts of oil at an average price of $1.45 per quart25 gallons of antifreeze at $6.50 per gallon24 tires at an average price of $49.75 per tire4 batteries at a total price of $140various auto supply accessories at a total price of

$245.75

A.) What was the total income from the sale of all these items?

B.) What percent of sales did the accessories represent?

C.) If Juan received a 5% commission on total sales, and a salary of $100 a week, how much did Juan earn that week?

D.) If Juan is offered another job for a salary of $150 a week without commission, should he quit his current job? Why or why not? Support your answer.

HSPA Prep Lesson 3Operations on Matrices

row

nmrows

mnmmm

n

n

n

aaaa

a

a

a

aaa

aaa

aaa

A

321

3

2

1

333231

232221

131211

A matrix is a rectangular array of numbers. We subscript entries to tell their location in the array

Matrices are identified by

their size.

Matrices and Row Operations

""

""

columnthj

rowthiaij

14

51 20513

3

1

6

2

0974

9852

7531

4212

44

5Barnett/Ziegler/Byleen Finite Mathematics 11e

Example: AdditionSolution

Add the matrices

Solution: First note that each matrix has dimensions of 3x3, so we are able to perform the addition. The result is shown at right:

Adding corresponding entries, we have

4 3 1 1 2 3

0 5 2 6 7 9

5 6 0 0 4 8

3 1 4

6 2 7

5 10 8

6

Example: SubtractionSolution

Now, we will subtract the same two matrices

Subtract corresponding entries as follows:

4 3 1 1 2 3

0 5 2 6 7 9

5 6 0 0 4 8

4 ( 1) 3 2 1 3

0 6 5 ( 7) 2 9

5 0 6 ( 4) 0 8

5 5 2

6 12 11

5 2 8

=

Matrix Addition

2 4 1 0

5 0 2 1

1 3 3 3

Example:

3 4

7 1

2 0

Scalar Multiplication

To do this, multiply each entry in the matrix by the number outside (called the scalar). This is like distributing a number to a polynomial.

Scalar Multiplication

2 4

4 5 0

1 3

Example:

8 16

20 0

4 12

To add matrices, we add the corresponding elements. They must have the same dimensions.

5 0 6 3

4 1 2 3A B

A + B

5 6 0 3

4 2 1 3

1 3

6 4

2 1 3 0 0 02.)

1 0 1 0 0 0

2 1 3

1 0 1

When a zero matrix is added to another matrix of the same dimension, that same matrix is obtained.

To subtract matrices, we subtract the corresponding elements. The matrices must have the same dimensions.1 2 1 1

3.) 2 0 1 3

3 1 2 3

1 1 2 ( 1)

2 1 0 3

3 2 1 3

0 3

3 3

5 4

4 1 6 51.)

6 3 7 3

1 3 2 2 1 52.)

4 0 5 6 4 3

2 6

13 0

1 4 7

2 4 8

ADDITIVE INVERSE OF A MATRIX:1 0 2

3 1 5A

1 0 2

3 1 5A

Find the additive inverse:

2 1 5

6 4 3

2 1 5

6 4 3

Scalar Multiplication:

1 2 3

1 2 3

4 5 6

k

We multiply each # inside our matrix by k.

1 2 3

1 2 3

4 5 6

k k k

k k k

k k k

3 01.) 3

4 5

9 0

12 15

2

1 2

2.) 5 4 1

0 5

x

y

x

2

5 10 5

20 5 5

0 25 5

x

y

x

18Barnett/Ziegler/Byleen Finite Mathematics 11e

Matrix EquationsExample: Find a, b, c, and d so that

2 1 4 3

5 6 2 4

a b

c d

Solution: Subtract the matrices on the left side:

2 1 4 3

5 6 2 4

a b

c d

Use the definition of equality to change this matrix equation into 4 real number equations:

a - 2 = 4 b + 1 = 3 c + 5 = -2 d - 6 = 4 a = 6 b = 2 c = -7 d = 10

On your own:

Practice Open-Ended (10 mins)

Discounts & Sales Tax

Sports Authority is having a sale. Every item in the store is discounted, however the discounts vary throughout the store!

1.) An jacket in the women’s clothing department is regularly priced at $60 and is on sale for $42.

Determine the percent discount rate for this

item.

2.) Find the total cost for a Nike tennis racket that is regularly priced at $125 and is on sale for

15% off. (*include a 6% sales tax.)

3.) If all of the sneakers are on sale at the % discount rate in #2, and the sales price for a pair of Nike sneakers is $85 (not including tax), what was the original price of the sneakers?