Equations and
Inequalities in
One Variable
Copyright © Cengage Learning. All rights reserved.
2
SECTION 2.2
Formulas
Copyright © Cengage Learning. All rights reserved.
Objectives
A
Solve a formula with numerical replacements for
all but one of its variables.
B
Solve formulas for the indicated variable.
C
Solve basic percent problems by translating
them into equations.
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A
Solving Formulas with Given
Values
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Solving Formulas with Given Values
Some formulas are probably already familiar to you—for
example, the formula for the area A of a rectangle with
length l and width w is A = lw.
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Example 1
Find y when x is 4 in the formula 3x – 4y = 2.
Solution:
We substitute 4 for x in the formula and then solve for y.
When
the formula
becomes
x=4
3x – 4y = 2
3(4) – 4y = 2
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Example 1 – Solution
cont’d
12 – 4y = 2
–4y = –10
y=
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Solving Formulas with Given Values
Note that in the last line of Example 1 we divided each side
of the equation by –4.
Remember that this is equivalent to multiplying each side of
the equation by
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B
Solving Formulas for an
Indicated Variable
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Example 4
Given the formula P = 2w + 2l, solve for w.
Solution:
To solve for w, we must isolate it on one side of the
equation.
We can accomplish this if we delete the 2l term and the
coefficient 2 from the right side of the equation.
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Example 4 – Solution
cont’d
To begin, we add –2l to both sides.
P + (–2l ) = 2w + 2l + (–2l )
P – 2l = 2w
To delete the 2 from the right side, we can multiply both
sides by .
(P – 2l ) =
(2w)
=w
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Example 4 – Solution
cont’d
The two formulas
P = 2w + 2l
and
give the relationship between P, l, and w.
They look different, but they both say the same thing about
P, l, and w.
The first formula gives P in terms of l and w, and the
second formula gives w in terms of P and l.
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Solving Formulas for an Indicated Variable
Rate Equation and Average Speed
Now we will look at a problem that uses what is called the
rate equation.
You use this equation on an intuitive level when you are
estimating how long it will take you to drive long distances.
For example, if you drive at 50 miles per hour for 2 hours,
you will travel 100 miles. Here is the rate equation:
Distance = rate time, or d = r t
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Solving Formulas for an Indicated Variable
The rate equation has two equivalent forms, one of which is
obtained by solving for r, while the other is obtained by
solving for t.
Here they are:
and
The rate in this equation is also referred to as average
speed.
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Solving Formulas for an Indicated Variable
The average speed of a moving object is defined to be the
ratio of distance to time.
If you drive your car for 5 hours and travel a distance of
200 miles, then your average rate of speed is
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Example 5
The first Ferris wheel was designed and built by
George Ferris in 1893. The diameter of the wheel was
250 feet. It had 36 carriages, equally spaced around the
wheel, each of which held a maximum of 40 people. One
trip around the wheel took 20 minutes. Find the average
speed of a rider on the first Ferris wheel.
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Example 5 – Solution
We can use 3.14 as an approximation for π. The distance
traveled is the circumference of the wheel, which is
C = 250π
≈ 250(3.14)
= 785 feet
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Example 5 – Solution
cont’d
To find the average speed, we divide the distance traveled
by the amount of time it took to go once around the wheel.
r =
=
=
39.3 feet per minute (to the nearest tenth)
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C
Basic Percent Problems
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Basic Percent Problems
The next examples in this section show how basic percent
problems can be translated directly into equations.To
understand these examples, we must recall that percent
means “per hundred.”
That is, 75% is the same as
fraction form.
, 0.75, and,
in reduced
Likewise, the decimal 0.25 is equivalent to 25%.
To change a decimal to a percent, we move the decimal
point two places to the right and write the % symbol.
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Basic Percent Problems
To change from a percent to a decimal, we drop the %
symbol and move the decimal point two places to the left.
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Basic Percent Problems
The table that follows gives some of the most commonly
used fractions and decimals and their equivalent percents.
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Example 9
What number is 15% of 63?
Solution:
To solve a problem like this, we let x equal the number in
question and then translate the sentence directly into an
equation.
Here is how it is done:
What number is 15% of 63?
x = 0.15 63
= 9.45
The number 9.45 is 15% of 63.
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