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 $y = 124$ when $x = 12$, find $y$ when $x = - 24$?

Answer 1

The inverse variation equation that relates $x$ and $y$ when you assume that $y$ varies inversely as $x$ is given by $xy = k$ where $k$ is constant. Use this information to evaluate the value of constant $k$ to obtain the particular inverse equation for the given condition.

Complete step by step solution:
The inverse variation equation $xy = k$ relates $x$ and $y$ such that $y$ varies inversely with $x$, here $k$ is a constant.

It is given in the question that $x$ has a value of $12$, when $y$ has a value of $124$.
Use this information to evaluate the value of constant $k$ to obtain the particular inverse equation for the given condition as shown below.

Substitute $x$ as $12$ and $y$ as $124$ in $xy = k$ and obtain the value of constant $k$ as follows:
$xy = k$
$\Rightarrow 12 \cdot 124 = k$
$\Rightarrow 1488 = k$

Therefore, the particular equation for the given situation is $xy = 1488$.

Now, obtain the value of $y$ when $x = - 24$ for the equation $xy = 1488$ as follow:

Substitute $x$ as $- 24$ in the equation $xy = 1488$ and solve for $y$ as shown below.
$\left( { - 24} \right)y = 1488$
$\Rightarrow \left( { - 24} \right)y = 1488$

Divide both sides by $- 24$ as shown below.
$\Rightarrow y = \dfrac{{1488}}{{ - 24}}$

Simplify the fraction and obtain the value of $y$ as shown below.
$\Rightarrow y = \dfrac{{1488}}{{ - 24}}$
$\Rightarrow y = - 62$

Therefore, the value of $y$ is $- 62$ for $x$ equal to $- 24$ for the inverse relation variation that relates $x$ and $y$ when you assume that $y$ varies inversely as $x$ given that if $y = 124$ when $x = 12$.

Note: The other way to obtain the value for $y$ when $x$ equal to $- 24$ in a inverse relation where $y = 124$ when $x = 12$.

The inverse relation $xy = k$ implies that ${x_1}{y_1} = {x_2}{y_2}$

Substitute ${x_1}$ as 12, ${y_1}$ as $124$ and ${x_2}$ as $- 24$ and obtain the value for ${y_2}$ as shown below.
$\left( {12} \right)\left( {124} \right) = \left( { - 24} \right){y_2}$
$\Rightarrow 1488 = \left( { - 24} \right){y_2}$

Divide both sides of the equation by $- 24$ as shown below.
$\Rightarrow \dfrac{{1488}}{{ - 24}} = {y_2}$
$\Rightarrow {y_2} = - 62$

Thus, the value of $y$ is $- 62$ for $x$ equal to $- 24$ for the inverse relation variation that relates $x$ and $y$ when you assume that $y$ varies inversely as $x$ given that if $y = 124$ when $x = 12$.

Always solve these types of questions very carefully as you can confuse between direct variation and inverse variation. In direct variation the relation is $y = kx$ and in inverse variation the relation is $y = \dfrac{k}{x}$.