1964 AMC 12 Problem 11

Attempt Problem 11 of the 1964 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 1964 AMC 12 solutions, or check the answer key.

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

11.

Given

2^x=8^{y+1}

and

9^y=3^{x-9},

find the value of

x+y.

18

21

24

27

30

Rewrite

8

2^3

and

9

3^2

Equate exponents to obtain two linear equations in

x

and

y

Writing both equations with common bases gives

\begin{aligned} x&=3y+3,\\ 2y&=x-9. \end{aligned}

Substitution yields

y=6

and

x=21,

x+y=27.

Therefore, the correct answer is D.

Problem 11 in other years