Beginning Algebra, 11th Edition

476 CHAPTER 7 Rational Expressions and Applications

Using Direct Variation

Suppose yvaries directly as x, and when Find ywhen

Since yvaries directly as x, there is a constant ksuch that We know that

when We substitute these values into and solve for k.

Equation for direct variation

Substitute the given values.

Constant of variation

Since and we have the following.

Let

Now find the value of ywhen.

Let

Thus, when y= 45 x=9. NOW TRY

y= 5 x= 5 # 9 = 45 x=9.

x= 9

y= 5 x k=5.

y=kx k=5,

k= 5

20 =k# 4

y=kx

y= 20 x=4. y=kx

y=kx.

y= 20 x=4. x=9.

EXAMPLE 1

Solving a Variation Problem

Step 1 Write the variation equation.

Step 2 Substitute the appropriate given values and solve for k.

Step 3 Rewrite the variation equation with the value of kfrom Step 2.

Step 4 Substitute the remaining values, solve for the unknown, and find the

required answer.

NOW TRY
EXERCISE 1
If Wvaries directly as r, and
when , find W
when .r= 10

W= 40 r= 5

NOW TRY ANSWER

1. 80

The direct variation equation is a linear equation. However, other kinds of

variation involve other types of equations.

y=kx

Direct Variation as a Power

yvaries directly as the nth power of xif there exists a real number ksuch that

the following is true.

ykxn

r

a  r^2

FIGURE 2

An example of direct variation as a power is the formula for the area of a circle,

Here, is the constant of variation, and the area varies directly as the

square of the radius. See FIGURE 2.

a=pr^2. p

Also, yis said to be proportional to x.The constant kin the equation for direct

variation is a numerical value, such as 3.00 in the gasoline price discussion. This

value is called the constant of variation.

Direct Variation

yvaries directly as xif there exists a constant ksuch that the following is true.

ykx

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