Question

Assume y varies inversely as x, how do you write the equation $y=6$ when $x=-3$?

Answer 1

Since y is inversely proportional to x, as given in the above question, the product of x and y will be a constant. So we can let the relation between them be $xy=c$. In the above question, we have also been given the values of the variables x and y which satisfies the given relation as $y=6$ when $x=-3$. On substituting the given values into the assumed relation $xy=c$, we will get the value of the constant c and hence the equation will be obtained.

Complete step by step solution:
We know that the inverse proportionality between two quantities or the variables means that their product is a constant. In the above question, it is given that y varies inversely as x. Therefore the product of x and y will be a constant and can be equated to some arbitrary constant, say c, so that we can write the equation
$\Rightarrow xy=c......\left( i \right)$
Now, according to the above question, we have $y=6$ when $x=-3$. Therefore, we can substitute $y=6$ and $x=-3$ in the above equation to get
$\begin{aligned} & \Rightarrow \left( 6 \right)\left( -3 \right)=c \\\ & \Rightarrow c=-18 \\\ \end{aligned}$
Therefore, the product of x and y is equal to $-18$. Now, we can substitute the value $c=-18$ into our assumed equation (i) to get
$\Rightarrow xy=-18$
Dividing both sides by x, we get
$\Rightarrow y=-\dfrac{18}{x}$
Hence, we have obtained the equation as $y=-\dfrac{18}{x}$.

Note:
We must not be confused between the term “direct proportion’ and “inverse proportion”. The direct proportion between two variables means that their ratio is a constant. And the inverse proportion means that their product is a constant.