HSPA Prep Lesson 3

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Transcript HSPA Prep Lesson 3

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 quart
25 gallons of antifreeze at $6.50 per gallon
24 tires at an average price of $49.75 per tire
4 batteries at a total price of $140
various 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 3
Operations on Matrices
Matrices and Row Operations
aij " i th row"
" j th column"
A matrix is a rectangular array of numbers. We
subscript entries to tell their location in the array
rows
a
a
a

11
12
13
row
a
a
a
21
22
23

A  a31 a32 a33


 
am1 am 2 am3
m n
a1n 

 a2 n 
 a3n 

 
 amn 

Matrices are
identified by
their size.
1 5 3 1 5 0 2
4 1
 2
6 
 
1 
 
  3
4 4
 2  1  2 4
 1 3

5
7


 2 5  8 9 


7
9 0
4
Example: Addition
Solution
 Add the matrices
 4 3 1   1 2 3
0 5 2   6 7 9

 

5 6 0   0 4 8
 Solution: First note that
each matrix has dimensions
of 3x3, so we are able to
perform the addition. The
result is shown at right:
5
 Adding corresponding
entries, we have
3 1 4 
6 2 7 


5 10 8 
Barnett/Ziegler/Byleen Finite
Mathematics 11e
Example: Subtraction
Solution
 Now, we will subtract the
same two matrices
 4 3 1   1 2 3
0 5 2   6 7 9

 

5 6 0   0 4 8
 Subtract corresponding
entries as follows:
3  2
1 3 
 4  (1)
 06

5

(

7)

2

9


 5  0
6  (4) 0  8 
=
6
 5 5 2 
 6 12 11


 5 2 8 
Matrix Addition
Example:
 2 4  1 0 
5 0    2 1  

 

1 3  3 3
 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
Example:
 2 4  8 16 




4 5 0   20 0


1 3  4 12 
To add matrices, we add the
corresponding elements. They must
have the same dimensions.
 5 0
A

 4 1
 5  6 0  3
A+B

 4  2 1 3 
6 3
B

2 3 
 1 3 


6 4 
 2 1 3 
0
2.) 



1 0 1
0
0 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
To subtract matrices, we subtract the
corresponding elements. The matrices
must have the same dimensions.
1

3.)  2
 3
 1 1
 2  1

 3  2
2

0
1

1
1

 2
2  (1)   0


0  3    3
1  3   5
1

3
3 
3

3
4
4 1 6 5
1.) 




6 3  7 3 
 2 6 


13 0 
1 3 2
 2 1 5   1 4 7 

2.) 
 



4 0 5 
6 4 3  2 4 8 
ADDITIVE INVERSE OF A MATRIX:
1
A
3
0 2

1 5 
 1
A  

3

0 2 

1 5
Find the additive
inverse:
 2 1 5   2 1 5

6 4 3  6 4 3 


 
Scalar
Multiplication:
2k 3k 
1 2 3
 1k




k  1 2 3  1k 2k 3k


 4 5 6
 4k 5k 6k 
We multiply each # inside our matrix
by k.
 3 0  9 0 

1.) 3 


 4 5 12 15
1 2 x   5 10 5 x 




2.) 5  4 y 1   20 5 y
5


2
0 5 x   0 25 5 x 2 
Matrix Equations
Example: Find a, b, c, and d so that
 a b   2 1  4 3
 c d    5 6    2 4

 
 

Solution: Subtract the matrices on the left side:
 a  2 b  1   4 3
 c  5 d  6   2 4

 

Use the definition of equality to change this matrix equation into 4 real
number equations:
a - 2 = 4 b + 1 = 3c + 5 = -2
c = -7
d = 10
18
d-6=4
a=6
b=2
Barnett/Ziegler/Byleen Finite
Mathematics 11e
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?