Equivalent Expressions Question 9

For all distinct nonzero values of $x$ and $y$ , which of the following expressions is equivalent to $\displaystyle \frac{\frac{x}{y}-1}{\frac{1}{y}-\frac{1}{x}}$ ?

1

x

\dfrac{1}{x}

\dfrac{1}{y}

\dfrac{x^2-1}{x-y}

Approach

Let's evaluate the numerator and denominator seperately, then combine them:

Numerator

\frac{x}{y}-1

=\frac{x}{y} - \frac{y}{y}

=\frac{x-y}{y}

Denominator

\frac{1}{y}-\frac{1}{x}

=\frac{x}{xy}-\frac{y}{xy}

= \frac{x-y}{xy}

\frac{x-y}{y}\div \frac{x-y}{xy}

= \frac{x-y}{y} \times \frac{xy}{x-y}

= \frac{xy}{y} = \boxed{x}