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Variation Solutions

• The constant of variation is the value    k    that relates    y    directly or inversely proportional to    x   .

Note

The value k can also relate y jointly or combinedly proportional to x, which will be discussed later in this lesson.

	Direct Variation	Inverse Variation
Equation in terms of y	$y=k{x}^n, n>0$	$y=\frac{k}{x^n}, n>0$
Equation in words	y varies directly as $x^n$	y varies inversely as $x^n$
Equation in terms of k	$k=\frac{y}{x^n}$	$k=x^{n}y$

• The algebraic    equation    needed depends on the    information    provided in the problem you are solving.

Note

Some texts may have $y=k{x}^n$ as joint variation due to repeated multiplication.

Example 1

Fill in the blanks.

1. For the given equation$y=\frac{\sqrt{6}}{x^2},$ y varies    inversely    as $x^2,$ and the constant of variation is $\underline{ \sqrt{\mathbf{6}} }.$

2. For the given equation$y=\mathrm{\pi }{x}^3,$ y varies    directly    as $\underline{ {\mathit{x}}^{\mathbf{3}} },$ and the constant of variation is    π   .

Example 2

Write the equation given the words.