Question

How do you find the variation constant and an equation of variation where y varies directly as x and $y=-2$ when $x=14$?

Answer 1

Hint: We first try to form the proportionality equation 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 answer:
The inversely proportional number is actually directly proportional to the inverse of the given number.
It’s given y varies directly with x which gives $y\propto x$.
To get rid of the proportionality we use the proportionality constant or variation constant which gives $y=kx$. This equation is the equation of variation.
Here, the number k is the proportionality constant. It’s given $y=-2$ when $x=14$.
We put the values in the equation $y=kx$ to find the value of k.
So, $-2=k\times 14$. Simplifying we get the value of k as
\[\begin{align}
  & -2=k\times 14 \\
 & \Rightarrow k=\dfrac{-2}{14}=-\dfrac{1}{7} \\
\end{align}\]
Therefore, the equation becomes with the value of k as $y=-\dfrac{1}{7}x$.
Now we simplify the equation to get
$\begin{align}
  & y=-\dfrac{1}{7}x \\
 & \Rightarrow 7y=-x \\
 & \Rightarrow x+7y=0 \\
\end{align}$
Then the equation for the variables is $x+7y=0$.

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.