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Direct and Inverse Variation

• Direct Variation

When y = k x for a nonzero constant k, we say that:

1. y varies directly as x, or2. y is proportional to x

The constant k is called the:

1. constant of variation, or2. constant of proportionality

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• Example 1

When a weight is attached to a spring, the distance the spring stretches varies directly as the weight. If a weight of 5 pounds stretches the spring 10 inches, find the distance the spring will be stretched with a 7 pound weight.

Let d = the distance the spring stretches w = the weight in pounds

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1) Write the variation equation

d kw

2) Use the given values to solve for k

10 5k

2k

3) Re-write the variation equation using the value of k

2d w

4) Determine the distance for the given weight

2 7d 14d

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• Conclusion The spring will travel a distance of 14 inches with the 7 pound weight.

• Note: In the previous problem, you could have easily solved it in your head. The steps that were used are very important for future work when the problems are more difficult and/or contain other types of variation.

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• Inverse Variation

When y = k/x for a nonzero constant k, we say that:

1. y varies inversely as x, or2. y is inversely proportional to x

The constant k is again called the:

1. constant of variation, or2. constant of proportionality

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• Example 2

Assume that y varies inversely as x and y = 3 when x = 12. Determine the value of y when x = 16

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1) Write the inverse variation equation

ky

x

2) Use the given values to solve for k

312

k

36k

3) Re-write the inverse variation equation using the value of k

36y

x

4) Determine the value of y for the given value of x

36

16y

9

4y

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