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)
06
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?