If (x)/(x-1)=(y^2+2y-1)/(y^2+2y-2), then x equals

1981 AMC 12 Problem 6

Attempt Problem 6 of the 1981 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 1981 AMC 12 solutions, or check the answer key.

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\frac{x}{x-1}=\frac{y^2+2y-1}{y^2+2y-2},

then

x

equals

y^2+2y-1

y^2+2y-2

y^2+2y+2

y^2+2y+1

-y^2-2y+1

Notice that each denominator is one less than its numerator

Set

A=y^2+2y-1

and compare

\frac{x}{x-1}

with

\frac A{A-1}

Let

A=y^2+2y-1.

The equation becomes

\frac{x}{x-1}=\frac{A}{A-1}.

Cross-multiplication gives

xA-x=Ax-A,

x=A=y^2+2y-1.

Therefore, the correct answer is A.