Question

If y varies inversely with x, and x = -10 when y = 60. How do you find the inverse variation equation and use it to find the value of y when x = 15?

Answer 1

Hint:The statement: y varies inversely as x means that when x increases, y decreases by the same factor. In other words, the expression \[xy\] is constant, to solve the inverse variation questions, we must know the equation \[y = \dfrac{k}{x}\], in which the we need to find the value of k by considering the values of x and y, then find out the inverse variation of y when x = 15.

Formula used:
\[y = \dfrac{k}{x}\]
In which, $k$ is the constant of variation and $x$ and $y$ are the variables.

Complete step by step answer:
Let us write the given data:
$y$ varies inversely with $x$,
$x$ = -10 when $y$ = 60
Hence, we need to find:
$x$ = 15 when $y$ =?
We know that the inverse variation sums are solved using the equation:
\[y\alpha k \times \dfrac{1}{x}\]
\[ \Rightarrow \]\[y = \dfrac{k}{x}\]
Given condition that y varies inversely with x, and x = -10 when y = 60 i.e.,
\[\left( {x,y} \right) \to \left( { - 10,60} \right)\]
Hence, by substitution we have:
\[y = \dfrac{k}{x}\]
\[60 = \dfrac{k}{{ - 10}}\]
Now, multiply both sides by -10 as
\[60\left( { - 10} \right) = k \times \dfrac{{ - 10}}{{ - 10}}\]
We know that,
\[\dfrac{{ - 10}}{{ - 10}} = + 1\]
Hence, simplifying the terms we get
\[60\left( { - 10} \right) = k \times \dfrac{{ - 10}}{{ - 10}}\]
\[ \Rightarrow \]\[ - 600 = k\]
As we have:\[y = \dfrac{k}{x}\]
Hence, substituting the value of k we get the value of y as:
\[y = - \dfrac{{600}}{x}\]
As, it is mentioned in the question that x = 15, then we have
\[y = - \dfrac{{600}}{{15}}\]
\[\therefore y = - 40\]

Therefore, the value of y when x = 15 is \[y = - 40\].

Note: Inverse variation equation are solved using the equation \[y = \dfrac{k}{x}\], read the problem carefully to determine if there are any other changes in the inverse variation equation, such as squares, cubes, or square roots. Use the information given in the problem to find the value of k, called the constant of variation or the constant of proportionality.