Question

Suppose y varies inversely with x. How do you write an equation for the inverse variation for $y=5$ when $x=-5$?

Answer 1

Hint: We first try to form an inverse relation for the variables. We take an arbitrary constant. We use the given values of the variables to find the value of the constant. Finally, we put the constant’s value to find the equation.

Complete step-by-step solution:

We have been given the relation between two variables $x$ and $y$ where y varies inversely with x.
The inversely proportional number is actually directly proportional to the inverse of the given number.
It’s given y varies inversely with x which gives $y\propto \dfrac{1}{x}$.
To get rid of the proportionality we use the proportionality constant which gives $y=\dfrac{k}{x}$.
Multiplying with $x$ on both sides we get $xy=k$.
Here, the number k is the proportionality constant.
It’s given $y=5$ when $x=-5$.
We put the values in the equation $xy=k$ to find the value of k.
So, $5\times \left( -5 \right)=k$. Simplifying we get
\[\begin{align}
  & 5\times \left( -5 \right)=k \\
 & \Rightarrow k=5\times \left( -5 \right)=-25 \\
\end{align}\]
Therefore, the equation becomes with the value of k as $xy=-25$.

Note: In a direct proportion, the ratio between matching quantities stays the same if they are divided. They form equivalent fractions. In an indirect (or inverse) proportion, as one quantity increases, the other decreases. In an inverse proportion, the product of the matching quantities stays the same.