For all nonzero numbers x and y such that x=(1)/(y), (x-1x)(y+1y) equals

1978 AMC 12 Problem 3

Attempt Problem 3 of the 1978 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1978 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

For all nonzero numbers

x

and

y

such that

x=\frac{1}{y},

\left(x-\frac1x\right)\left(y+\frac1y\right)

equals

2x^2

2y^2

x^2+y^2

x^2-y^2

y^2-x^2

Use the reciprocal relation to replace

\frac{1}{x}

and

\frac{1}{y}

The two factors become

x-y

and

x+y

Since

x=\frac{1}{y},

we also have

y=\frac{1}{x}.

Thus the product is

(x-y)(y+x)=x^2-y^2.

Therefore, the correct answer is D.