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

Hint: 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 \[y = \dfrac{k}{x}\].