Define the operation “ ∘ ” by x∘ y=4x-3y+xy, for all real numbers x and y. For how many real numbers y does 3∘ y=12?

1993 AMC 12 Problem 4

Attempt Problem 4 of the 1993 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 1993 AMC 12 solutions, or check the answer key.

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Define the operation “

\circ

” by

x\circ y=4x-3y+xy,

for all real numbers

x

and

y.

For how many real numbers

y

does

3\circ y=12?

0

1

3

4

more than

4

Substitute

x=3

into the operation

Check whether the terms containing

y

cancel

For every real

y,

3\circ y=4(3)-3y+3y=12.

Hence every real number

y

works, so there are more than

4

such numbers. Thus the correct answer is E.