# Assume that y varies inversely as x. If y = -6 when x = -2, how do you find y when x = 5?

Apr 5, 2016

At $x = 5 \text{; } y = \frac{12}{5} = 2 \frac{2}{5}$

#### Explanation:

This is a version ratios (ratios turn up all over the place).

Consider the example: $\frac{a}{b} = \frac{c}{d} \implies a = b \times \frac{c}{d}$

This is exactly what we have in this question.

Let $k$ be a constant

$y = k \times \frac{1}{x}$

$\textcolor{g r e e n}{y = \frac{k}{x}}$ ...................................(1)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the value of } k}$

Given condition: at $y = - 6 \text{ } x = - 2$

Substituting in equation (1)

$\text{ } - 6 = \frac{k}{- 2}$

Multiply both sides by 2

$\left(- 6\right) \times 2 = k \times \frac{2}{- 2}$

But $\frac{2}{- 2} = - 1$

$\implies - 12 = - k$

So $k = + 12$

Thus equation (1) becomes

$\textcolor{g r e e n}{y = \frac{12}{x}}$.................................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So at $x = 5$ we have:

$\text{ } y = \frac{12}{5} = 2 \frac{2}{5}$