Integrated Algebra
NOTES: Solving word problems using a system
Name
Now that you can solve equations with two variables, we are going to practice using
a system to solve word problems.
EX:
Suppose a paper manufacturer publishes a newsletter. Expenses are $0.90 for
printing and mailing each copy, plus $600 for research and writing. The price of
the newsletter is $1.50 per copy. How many copies of the newsletter does the
company need to sell to break even?
Key: (independent) x = number of copies
(dependent) y = money (Two types; income and expenses)
System: income
expenses
:
y = 1.50x
:
y = 0.90x + 600
Solution ; (substitution)
y = 1.5x
y = .9x + 600
1.5x = .9x + 600
.6x = 600
x = 1000
The company needs to sell 1000 copies.
EX: Suppose you invest $10,000 in equipment to manufacture a new board game.
Each game costs $2.65 in materials to manufacture and you plan to sell them for
$20. How many games must you make to break even (assume that all you make do
sell) ?
Key: x = number of games
y = money
(independent)
(dependent)
System: Ask yourself: what two relationships are there with the money?
Income (y): 20x
Expenses (y) : 2.65 + 10000
Substitute:
System:
y = 20x
y = 2.65x + 10000
20x = 2.65x + 10000
17.35x = 10000
x = 576.3688 … so you need to sell 577 games.
Integrated Algebra
TRY:
1. The costs of scripts for the spring musical is $254. The cost of props and
costumes is $400. You must pay royalty fees of $1.20 per ticket to the play’s publisher.
You charge $4 per ticket. You estimate that you will make $150 from refreshments.
How many tickets must you sell to break even?
2. Suppose you are going into business mowing lawns. You purchased a lawn mower
for $300. Gas and oil costs you $4.00 per lawn. You are charging $35 per lawn. How
many lawns must you mow to break even?
Integrated Algebra
Word problems that we solved earlier in one variable may be easier using two.
EX: The difference of two supplementary angles is 36˚. Find both angles.
SINGLE VARIABLE
TWO VARIABLES (system)
Key: 1st angle = x
2nd angel = 180 – x
1st angle = x
2nd angle = y
Equation: x – (180 – x) = 36
System: x + y = 180
x - y = 36
x – 180 + x = 36
2x – 180 = 36
2x = 216
X = 108
2x
1st = 108˚
2nd = 72˚
= 216
x =108
1st = 108˚
2nd = 72˚
EX: You have 10 coins that total $0.85. They are dimes and nickels. How many of each do
you have?
SINGLE VARIABLE
Dimes
Nickels
x
10 - x
.10x
.05(10-x)
Equation: .10x + .05(10-x) = 0.85
.
TWO VARIABLES
=10x + 5(10-x) = 85
10x + 50 – 5x = 85
5x + 50 = 85
5x = 35
x=7
Dimes = 7
Nickels = 3
x = dimes
y = nickels
System:
x + y = 10
10x + 5y = 85
-5x – 5y = -50
10x + 5y = 85
5x
= 35
x=7
Dimes = 7
Nickels = 3
Integrated Algebra
TRY:
1.
You have a total of $2.55. You have a total of 15 dimes and quarters. How many
of each do you have?
2. A rectangle has a perimeter of 30 feet. The length is 3 feet more than the width.
Find the length and the width.
Integrated Algebra
EX : 2 popcorns and 3 sodas cost $4.75. One popcorn and 4 sodas cost $4.25
How much does one soda cost?
Key:
p = popcorn
s = soda
System: need two relationships:
Eliminate: 2p + 3s = 475
p + 4s = 425
2p + 3s = 475
p + 4s = 425
2p + 3s = 475
-2(p + 4s = 425)
2p + 3s = 475
-2p - 8s = -850
-5s= -375
s = 75
soda is $0.75 or 75¢
popcorn : p + 4(.75) = 4.25
p + 3.00 = 4.25
p = 1.25
popcorn is $1.25
EX: A company orders brass and steel parts. One shipment contained 3 brass and 10
steel parts and cost $48. Another shipment cost $54 and contained 7 brass and 4
steel parts. How much is one brass and one steel part?
Key: b = brass
s = steel
System: need two relationships : 3b + 10s = 48
7b + 4s = 54
Eliminate, use LCM of s: +20 and -20
3b + 10s = 48
7b + 4s = 54
2(3b + 10s = 48)
-5(7b + 4s = 54)
Solve for s: 3(6) + 10s = 48
18 + 10s = 48
10s = 30
s=3
6b + 20s = 96
-35b – 20s = -270
-29b = 174
b= 6
Brass is $6.00
Steel is $3.00
Integrated Algebra
TRY:
1. At the ballpark 3 slices of pizza and 2 sodas cost $6. 4 slices of pizza
and 4 sodas cost $9. How much is a slice of pizza?
2.
At the movies, 2 soft pretzels and 3 sodas cost $7.25. 5 soft pretzels and 2 sodas
cost $11.25. How much did a pretzel cost?
Integrated Algebra
HOMEWORK: Solving word problems using a system
Name
Solve each word problem using a system of equations.
1. The perimeter of a rectangle is 24 feet. The length is five times the width. Find
the length and width.
2. At the bookstore, 4 packages of pens and 2 notebooks cost $8.20. One package of
pens and 3 notebooks cost $7.80. Find the cost of each item.
3. You are starting an office cleaning service. You have spend $550 on equipment. To
clean one office it costs you $6 worth of supplies. You are going to charge $35 per
office. How many offices will you need to clean to break even?
Integrated Algebra
4. You are taking a test worth 100 points. It is made up of a total of 40 questions.
Some are multiple choice questions and some are short answer questions. The
multiple choice questions are worth 2 points and the short answer are worth 4
points. How many of each type of question are on the test?
5. A collection of nickels and dimes is worth $2.25. There are 28 coins in the jar.
How many of each type are in the jar?
6. A sporting goods manufacturer has spent $2000 researching a new product. It will
cost $4.50 to manufacture the item. It will sell for $12. How many items do you
have to sell to break even?