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For integers x and y, \(3^{(4x+12)}=5^{(3x+y)}\). What is the value of y?
A. -12
B. -3
C. 0
D. 9
E. Cannot be determined
Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITIONGRE - Math BookThe key word in this question is INTEGERS
Notice that, if x is an integer, then 4x+12 is an integer, which means
3^(4x+12) will equal the product of a bunch of 3's Likewise, if x and y are integers, then 3x+y is an integer, which means
5^(3x+y) will equal the product of a bunch of 5's Given these conditions, it seems impossible that 3^(4x+12) could ever equal 5^(3x+y)
HOWEVER, if the exponents 4x+12 and 3x + y both equal ZERO, then we get 3^0 and 5^0, and both of these evaluate to equal 1 - PERFECT!
So, let 4x+12 = 0 and let 3x+y = 0
Now we'll solve this system of equations for x and y.
First, if 4x+12 = 0, then x = -3
If x = -3, then we can take 3x+y = 0 and replace x with -3 to get: 3(-3) + y = 0
Simplify: -9 + y = 0
Solve: y = 9
So, x = -3 and y = 9, is a solution to the equation 3^(4x+12) = 5^(3x+y)
Answer: D
Cheers,
Brent