9.3 Applications

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9.3 Applications

• FUN with Story Problems!

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Number problems

• The sum two integers is 6. Two times the smaller
  integer is 24 less
  
• than the larger integer. Find the integers.

• Label the unknowns using two variables
• Write a system of equations
• Solve by the easiest method

Let x smaller integer y bigger integer
x y 6
x y 6
2x y - 24
2x y - 24
3x - 18 x - 6

• 6 y 6
• y 12

The two numbers are 6 and 12.
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Strategy for solving Word Problems
in two unknowns
2. Assign variables to two unknown quantities
a) Assign x to one unknown quantity and
let y represent the other.
b) Find two relationships in terms of x and y.
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Perimeter
5. The perimeter of a rectangle is 72 cm. The sum
of two times the length and three times the w
idth is 88 cm. Find the dimensions.
Let l length w width
2l 2w 72
- 2l 2w - 72
2l 3w 88
2l 3w 88
w 16
2l 2(16) 72
2l 32 72 2l 40 l 20
The dimensions of the rectangle are 20 X 16.
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Money
9. A theater sold 520 worth of tickets. An adult
ticket cost 3 and a childs ticket cost 2.
If 190 tickets were sold how many tickets
of each kind were sold?
Let a adults tickets c childrens ti
ckets
Normally, one equation comes from the amount of
money.
3a 2c 520
3a 2c 520 -2a 2c - 380
a c 190
The other equation comes from the total number of

items there are.
a 140
140 c 190 c 50
There were 140 adults tickets and 50 childrens
tickets sold.
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Age
14. A father is three times as old as his son.
After twelve years his age will be two time
s the age of his son. Find their present ages.
Let f the fathers age s the sons age
f 12 fathers age in 12 years
s 12 sons age in 12 years
f 3s
f 12 2(s 12)
3s 12 2(s 12)
3s 12 2s 24 s 12 24 s 12
f 3(12) f 36
The son is 12 and the father is 36.
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Mixture Problems
25. A 30 dye solution is to be mixed with an 80
dye solution to make 60 L of 50 dye soluti
on. How many liters of each is needed?
25. Let x 30 dye solution y 80
dye solution
One equation represents the amount of mix x
y 60
One equation represents what is being mixed
.3x .8y .5(60)
x y 60 .3x .8y 30
- .3x - .3y -18
.3x .8y 30
.5y 12
y 24
x 36
You need 36L of 30 solution and 24L of 80
solution.
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Number Problem
The difference of two numbers is 27. The larger
number is five less than three times the other.
Find both numbers.
1. The difference is positive. ? The first
is the larger.
2. Assign x to the larger and y to the
smaller.
Write equations
Substitute
x 3(16) - 5
x 43
Use Substitution
(3y - 5) - y 27
2y 32
Check
43 - 16 27 and 43 3(16) - 5
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Interest Problem
Mrs. Smith has 20,000 to invest. If part is
invested at 12 and the rest at 8, how much
should be invested at each rate to yield 11 on
the total amount invested.
Interest problems can be set up in a MATRIX for
clarity
Part A Part B Total Invested
Amount x y
20,000
Rate 12 8
11
Yield .12x .08y
.11(20000)
Equations
Amount x y
20,000 Interest .12x .08y
.11(20000)
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Equations
Multiply by 100 (
)
Multiply by - 8 (
)
4x 60000
15000 y 20000
Replace x in 1
.12(15000) .08(5000) .11(20000) 2200
Check
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Coin Problem
A collection of dimes and quarters has a total
value of 2.20. If there are 3 times as many
dimes as quarters, how many of each is in the
collection.
If clause there are 3 times as many dimes as
quarters
Translate The number of dimes is 3 times the
number of quarters
Let x be the number of dimes and y the number
of quarters
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Coin Problem
Write equations
Use Substitution 10(3y) 25y 220
Combine like terms
Replace y in 1
Check
10(12) 25(4) 220 or 1.20 1.00 2.20
There were 12 dimes and 4 quarters in the
collection.