Question

How do you write an inverse variation equation that relates $x$ and $y$ when you assume that $y$ varies inversely as $x$ given that if $y=3.2$ when $x=-5.5$, find $y$ when $x=6.4$?

Answer 1

Hint: Inverse variation is a type of relationship that varies from direct variation because it defines a non-linear relationship between two variables. When one of two quantities has inverse variation, as one increases, the other decreases.
When driving to a specific location, for example, as the pace increases, the time it takes to arrive at that location decreases. The time it takes to arrive at that position increases as your speed decreases. As a result, the values are inversely proportional.

Complete step by step solution:
We represent inverse variation as $$xy=k$$
Inverse variation $1$ :
From the question, we know that $y=3.2$ and $x=-5.5$.
Since we have the values of $y$ and $x$ , we can find the inverse variation value which is $k$
$$\begin{gathered} k=x\times y \\ =(-5.5)\times 3.2 \\ =-17.6 \end{gathered}$$
The value of the constant of inverse variation, $k=-17.6$ when $y=3.2$ and $x=-5.5$

Inverse variation $2$ :
From the question, we know that $x=6.4$ and we have the value of the constant $k=-17.6$ also.
Since we have the values of $x$ and $k$ , we can find the value of $y$
$k=x\times y$
Since we have to find $y$ , we rewrite the equation as
$$\begin{gathered} y={\displaystyle \frac{k}{x}} \\ ={\displaystyle \frac{-17.6}{6.4}} \\ =-2.75 \end{gathered}$$
The value of $y=-2.75$ when $x=6.4$.

Note:
We can also solve the second part of the question by using the Product Rule of Inverse Variation.
It states that if $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are solutions of a particular inverse variation, then $x_1{y}_1=k$ and $x_2{y}_2=k$
On substituting $x_1{y}_1$ for $k$ , we get,
$x_1{y}_1=x_2{y}_2$ or ${\displaystyle \frac{x_1}{x_2}}={\displaystyle \frac{y_2}{y_1}}$
The equation $x_1{y}_1=x_2{y}_2$ is known as the product rule of inverse variation.