OG 13 - problem 41

gdshamain wrote:41. Is 4^x+y = 8^10?

(1) x-y = 9
(2) y/x = 1/4

When posting questions, please use brackets and spaces to avoid ambiguity.
Based on your solution, the question SHOULD read:

Does 4^(x+y) = 8^10?
1) x - y =9
2) y/x = 1/4

Cheers,
Brent

Does 4^(x+y) = 8^10?
1) x - y = 9
2) y/x = 1/4

This is a great candidate for rephrasing the target question.
Aside: We have a free video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Target question: Does 4^(x+y) = 8^10?
Take 4^(x+y) = 8^10 and rewrite each side with the same base of 2 to get:
(2^2)^(x+y) = (2^3)^10
Simplify to get: 2^(2x + 2y) = 2^30 [power of a power rule]
For this equation to hold true, it must be the case that 2x + 2y = 30
Divide both sides by 2 to get x + y = 15

We can now REPHRASE our target question...
REPHRASED target question: Does x + y = 15?

Statement 1: x - y = 9
There are several values of x and y that satisfy this condition. Here are two:
Case a: x = 12 and y = 3, in which case x + y = 15
Case b: x = 10 and y = 1, in which case x + y ≠ 15
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y/x = 1/4
Cross multiply to get: x = 4y
There are several values of x and y that satisfy this condition. Here are two:
Case a: x = 12 and y = 3, in which case x + y = 15
Case b: x = 8 and y = 2, in which case x + y ≠ 15
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x - y = 9
Statement 2 tells us that x = 4y
We now have a system of TWO linear equations with TWO variables, so we COULD easily solve this system for x and y, which means we COULD determine whether x + y = 15.
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent