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# For integers x and y,3^(4x+12)=5^(3x+y). What is the value of y?

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For integers x and y,3^(4x+12)=5^(3x+y). What is the value of y?  [#permalink]

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21 Feb 2017, 04:17
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12
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Difficulty:

65% (hard)

Question Stats:

49% (01:38) correct 51% (01:57) wrong based on 237 sessions

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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

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Re: For integers x and y,3^(4x+12)=5^(3x+y). What is the value of y?  [#permalink]

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21 Feb 2017, 04:43
10
4
Bunuel wrote:
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

3^(4x+12)=5^(3x+y)

This relation will hold true if the power of 3 and 5 on both sides of equation become zero because $${anything}^0 = 1$$

i.e. 4x+12 = 0 and 3x+y=0
i.e. x = -2 and y = 9

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Joined: 11 Sep 2015
Posts: 3243
Re: For integers x and y,3^(4x+12)=5^(3x+y). What is the value of y?  [#permalink]

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22 Feb 2017, 07:34
3
Top Contributor
1
Bunuel wrote:
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

The 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)

Cheers,
Brent
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Re: For integers x and y,3^(4x+12)=5^(3x+y). What is the value of y?  [#permalink]

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22 Feb 2017, 07:35
Top Contributor
GMATinsight wrote:

i.e. 4x+12 = 0 and 3x+y=0
i.e. x = -2 and y = 9

Small error with the x-value above. Should be x = -3

Cheers,
Brent
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Re: For integers x and y,3^(4x+12)=5^(3x+y). What is the value of y?  [#permalink]

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23 Feb 2017, 09:45
1
Bunuel wrote:
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

In order for the two sides of the equation to be equal, the exponent (4x+12) must equal zero and the exponent (3x+y) must equal zero, since 3^0 = 1 and 5^0 = 1. Thus:

4x + 12 = 0

4x = -12

x = -3

and

3x + y = 0

y = -3x

Since x = -3, y = -3(-3) = 9.

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Re: For integers x and y,3^(4x+12)=5^(3x+y). What is the value of y?  [#permalink]

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17 Jul 2017, 05:10
GMATinsight wrote:
Bunuel wrote:
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

3^(4x+12)=5^(3x+y)

This relation will hold true if the power of 3 and 5 on both sides of equation become zero because $${anything}^0 = 1$$

i.e. 4x+12 = 0 and 3x+y=0
i.e. x = -2 and y = 9

Hello GMATinsight - You solution is really the quickest way to do! Somehow I spent 4 mins to solve this.

Just one correction, the value you mentioned for x is incorrect, x should be -3
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Re: For integers x and y,3^(4x+12)=5^(3x+y). What is the value of y?  [#permalink]

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17 Jul 2017, 05:14
1
For integers x and y, $$3^{4x+12}=5^{3x+y}$$. What is the value of y?

$$3^{4x+12}=5^{3x+y}$$

This is only possible when both 3 and 5 are raised to the power of ZERO.

4x + 12 = 0

3x + y = 0

Solving these two equations we get:

$$x = -3$$

$$y = 9$$

_________________

"Nothing in this world can take the place of persistence. Talent will not: nothing is more common than unsuccessful men with talent. Genius will not; unrewarded genius is almost a proverb. Education will not: the world is full of educated derelicts. Persistence and determination alone are omnipotent."

Best AWA Template: https://gmatclub.com/forum/how-to-get-6-0-awa-my-guide-64327.html#p470475

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Posts: 2830
Re: For integers x and y,3^(4x+12)=5^(3x+y). What is the value of y?  [#permalink]

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18 Aug 2018, 17:25
Bunuel wrote:
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

Since 3^(4x+12) = 5^(3x+y) and 3 and 5 are relatively prime to each other (that is, they don’t have a common factor other than 1), the only way those two expressions can be equal is if each base is raised to the zero power and thus each side is equal to 1.

Thus:

4x + 12 = 0

4x = -12

x = -3

Substituting, we have:

3(-3) + y = 0

-9 + y = 0

y = 9

_________________

Jeffery Miller

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Re: For integers x and y,3^(4x+12)=5^(3x+y). What is the value of y? &nbs [#permalink] 18 Aug 2018, 17:25
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