# If y varies inversely with x, and x = -10 when y = 60. How do you find the inverse variation equation and use it to find the value of y when x = 15?

Feb 17, 2017

The equation is $y = - \frac{600}{x}$

At $x = 15$ the value of $y$ is - 40

#### Explanation:

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

Given condition $\left(x , y\right) \to \left(- 10 , 60\right)$

So by substitution we have:

$\textcolor{g r e e n}{y = \frac{k}{x} \text{ "->" } 60 = \frac{k}{- 10}}$

Multiply both sides by $\textcolor{red}{\left(- 10\right)}$

$\textcolor{g r e e n}{60 \textcolor{red}{\times \left(- 10\right)} = k \times \frac{\textcolor{red}{- 10}}{- 10}}$

But $\frac{- 10}{- 10} = + 1$

$- 600 = k$

Giving: $y = \frac{k}{x} \text{ "->" } y = - \frac{600}{x}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Suppose $x = 15$ as in the question. Then we have:

$y = - \frac{600}{15} = - 40$

Feb 17, 2017

$y = - 40$

#### Explanation:

In an inverse variation, as one quantity increases, the other one decreases.

$y$ varies inversely with $x$ can be written as: $\text{ } y \propto \frac{1}{x}$

A proportion can be made into an equation by using a constant:

$y \propto \frac{1}{x} \text{ "rarr" "y = color(red)(k)/x" "rarr" } \textcolor{red}{k} = x \times y$

Use the two values given to find the value of $k :$

$\textcolor{red}{k} = - 10 \times 60 = \textcolor{red}{- 600}$

The equation can therefore be written as: $\text{ } y = \frac{\textcolor{red}{- 600}}{x}$

Now we know the exact relationship between $x$ and $y$

If $x = 15 \text{ } y = - \frac{600}{15} = - 40$