Precalculus Examples

$y = 4$ ,

$x = - 4$ ,

$z = 2$

Step 1

When three variable quantities have a constant ratio, their relationship is called a direct variation. It is said that one variable varies directly as the other two vary. The formula for direct variation is

$y = k x z^{2}$ , where

$k$ is the constant of variation.

$y = k x z^{2}$

Step 2

Solve the equation for

$k$ , the constant of variation.

$k = \frac{y}{x z^{2}}$

Step 3

Replace the variables

$x$ ,

$y$ , and

$z$ with the actual values.

$k = \frac{4}{(- 4) \cdot {(2)}^{2}}$

Step 4

Cancel the common factor of

$4$ and

$- 4$ .

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Step 4.1

Factor

$4$ out of

$4$ .

$k = \frac{4 (1)}{(- 4) \cdot {(2)}^{2}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor

$4$ out of

$(- 4) \cdot {(2)}^{2}$ .

$k = \frac{4 (1)}{4 (- {(2)}^{2})}$

Step 4.2.2

Cancel the common factor.

$k = \frac{4 \cdot 1}{4 (- {(2)}^{2})}$

Step 4.2.3

Rewrite the expression.

$k = \frac{1}{- {(2)}^{2}}$

Step 5

Cancel the common factor of

$1$ and

$- 1$ .

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Step 5.1

Rewrite

$1$ as

$- 1 (- 1)$ .

$k = \frac{- 1 \cdot - 1}{- {(2)}^{2}}$

Step 5.2

Move the negative in front of the fraction.

$k = - \frac{1}{2^{2}}$

Step 6

Raise

$2$ to the power of

$2$ .

$k = - \frac{1}{4}$

Step 7

Write the equation of variation such that

$y = k x z^{2}$ , replacing

$k$ with the

$- \frac{1}{4}$ .

$y = - \frac{z^{2} x}{4}$

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