Question

Each pair of points $\left( {5.4,3} \right)$ and $\left( {2,y} \right)$ is on the graph of an inverse variation, how do you find the missing value?

Answer 1

In the question it is given that the given pair of values form an inverse function. We will first form an equation using the first variable, second variable and constant using the concept of inverse variation. Then in that equation we will put the first pair of values to get the value of the constant. Then we will form a general equation using the value of the constant. Then we will put the value of one of the numbers from the second pair to get the value of the missing number.

Complete step by step answer:
Since the given pair forms an inverse variation, so the first variable i.e., $x$ varies inversely with respect to the second variable i.e., $y$. It means when the first variable increases, the second variable decreases and vice-versa. Also, the product of the first and the second variable will be constant.
Mathematically,
$y \propto \dfrac{1}{x}$
Removing the proportionality sign we get;
$\Rightarrow xy = k$
Now, the first pair is $\left( {5.4,3} \right)$. So, putting the value we get;
$\Rightarrow 5.4 \times 3 = k$
Solving we get;
$\Rightarrow k = 16.2$
So, the general solution becomes;
$\Rightarrow xy = 16.2$
Now we will substitute the values from the second pair. So, we get;
$\Rightarrow 2y = 16.2$
Shifting we get;
$\Rightarrow y = \dfrac{{16.2}}{2}$
On dividing we get;
$\Rightarrow y = 8.1$
Hence the pair is $\left( {2,8.1} \right)$.

Note: One point to note here is that in the case of inverse variation the curve we get will be rectangular hyperbola. If we replace $k$ by ${c^2}$, we will get $xy = {c^2}$. This is the equation of a rectangular hyperbola. In case of direct variation we will get an equation of straight line.