Examples

y = 8

$y = 8$ ,

x = 2

$x = 2$ ,

z = - 7

$z = - 7$

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}

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

k

$k$ is the constant of variation.

y = k x z^{2}

$y = k x z^{2}$

Step 2

Solve the equation for

k

$k$ , the constant of variation.

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

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

Step 3

Replace the variables

x

$x$ ,

y

$y$ , and

z

$z$ with the actual values.

k = \frac{8}{(2) \cdot {(- 7)}^{2}}

$k = \frac{8}{(2) \cdot {(- 7)}^{2}}$

Step 4

Cancel the common factor of

8

$8$ and

2

$2$ .

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Step 4.1

Factor

2

$2$ out of

8

$8$ .

k = \frac{2 \cdot 4}{2 \cdot {(- 7)}^{2}}

$k = \frac{2 \cdot 4}{2 \cdot {(- 7)}^{2}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor

2

$2$ out of

2 \cdot {(- 7)}^{2}

$2 \cdot {(- 7)}^{2}$ .

k = \frac{2 \cdot 4}{2 ({(- 7)}^{2})}

$k = \frac{2 \cdot 4}{2 ({(- 7)}^{2})}$

Step 4.2.2

Cancel the common factor.

k = \frac{2 \cdot 4}{2 {(- 7)}^{2}}

$k = \frac{2 \cdot 4}{2 {(- 7)}^{2}}$

Step 4.2.3

Rewrite the expression.

k = \frac{4}{{(- 7)}^{2}}

$k = \frac{4}{{(- 7)}^{2}}$

k = \frac{4}{{(- 7)}^{2}}

$k = \frac{4}{{(- 7)}^{2}}$

k = \frac{4}{{(- 7)}^{2}}

$k = \frac{4}{{(- 7)}^{2}}$

Step 5

Raise

- 7

$- 7$ to the power of

2

$2$ .

k = \frac{4}{49}

$k = \frac{4}{49}$

Step 6

Write the equation of variation such that

y = k x z^{2}

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

k

$k$ with the

\frac{4}{49}

$\frac{4}{49}$ .

y = \frac{4 x z^{2}}{49}

$y = \frac{4 x z^{2}}{49}$

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