If x and y are nonzero numbers such that x=1+1y and y=1+1x, then y equals:

1968 AMC 12 Problem 14

Attempt Problem 14 of the 1968 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 1968 AMC 12 solutions, or check the answer key.

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14.

x

and

y

are nonzero numbers such that

x=1+\dfrac1y

and

y=1+\dfrac1x,

then

y

equals:

x-1

1-x

1+x

-x

x

Multiply each equation by its denominator

Both equations produce an expression for

xy

The equations give

xy=y+1

and

xy=x+1.

Therefore

x+1=y+1,

y=x.

Therefore, the correct answer is E.