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Chapter 9
Equations,
Inequalities and
Problem Solving
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
9.5
Formulas and Problem
Solving
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Formulas
A formula is an equation that states a known
relationship among multiple quantities (has more than
one variable in it)
A = lw
(Area of a rectangle = length · width)
I = PRT
(Simple Interest = Principal · Rate · Time)
P=a+b+c
(Perimeter of a triangle = side a + side b + side c)
d = rt
(distance = rate · time)
V = lwh
(Volume of a rectangular solid = length · width · height)
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
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Example
A flower bed is in the shape of a triangle with one side
twice the length of the shortest side, and the third side is 30
feet more than the length of the shortest side. Find the
dimensions if the perimeter is 102 feet.
The formula for the perimeter of a triangle is P = a + b + c.
If we let
x = the length of the shortest side, then
2x = the length of the second side, and
x + 30 = the length of the third side
continued
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
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continued
Formula: P = a + b + c
Substitute: 102 = x + 2x + x + 30
102 = x + 2x + x + 30
102 = 4x + 30
102 – 30 = 4x + 30 – 30
72 = 4x
72 4 x

4
4
18 = x
continued
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Martin-Gay, Developmental Mathematics, 2e
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continued
Check: If the shortest side of the triangle is 18 feet, then
the second side is 2(18) = 36 feet, and the third side is
18 + 30 = 48 feet. This gives a perimeter of
P = 18 + 36 + 48 = 102 feet, the correct perimeter.
State: The three sides of the triangle have a length of 18
feet, 36 feet, and 48 feet.
continued
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Martin-Gay, Developmental Mathematics, 2e
6
Solving a Formula for a Variable
It is often necessary to rewrite a formula so that it is
solved for one of the variables.
To solve a formula or an equation for a specified
variable, we use the same steps as for solving a
linear equation except that we treat the specified
variable as the only variable in the equation.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
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Solving Equations for a Specified
Variable
Step 1:Multiply on both sides to clear the equation of
fractions if they appear.
Step 2:Use the distributive property to remove
parentheses if they appear.
Step 3:Simplify each side of the equation by combining
like terms.
Step 4:Get all terms containing the specified variable
on one side and all other terms on the other side
by using the addition property of equality.
Step 5:Get the specified variable alone by using the
multiplication property of equality.
Martin-Gay, Developmental Mathematics, 2e
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Example
Solve T = mnr for n.
T
mnr

mr mr
T
n
mr
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
9
Example
Solve for A = PRT for T.
A  P  P  P  PRT
A  P  PRT
A  P PRT

PR
PR
A P
T
PR
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Martin-Gay, Developmental Mathematics, 2e
10
Example
Solve A  P  PRT for P.
A  P(1  RT )
Factor out P on the right side.
A
P(1  RT )

1  RT
1  RT
Divide both sides by 1 + RT.
A
P
1  RT
Simplify.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
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