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What is the value of x? (1) 8^(x + 2) = 16^(4 + y) (2) 3^(x − y) =27

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What is the value of x? (1) 8^(x + 2) = 16^(4 + y) (2) 3^(x − y) =27  [#permalink]

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New post 04 Apr 2018, 23:43
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A
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C
D
E

Difficulty:

  25% (medium)

Question Stats:

93% (01:32) correct 7% (01:02) wrong based on 47 sessions

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Re: What is the value of x? (1) 8^(x + 2) = 16^(4 + y) (2) 3^(x − y) =27  [#permalink]

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New post 05 Apr 2018, 00:40
1
Bunuel wrote:
What is the value of x?

(1) \(8^{(x + 2)} = 16^{(4 + y)}\)

(2) \(3^{(x − y)} =27^{(y + 6)}\)


Before explicitly solving, we'll first look for a property-based approach without calculation.
This is a Logical approach.

(1) , (2) Each of these gives only one equation with two variables and so has infinite solutions.
Insufficient.

Combined:
We can translate (1) and (2) into two distinct linear equations with two variables, which has one solution!
No need to actually calculate.

(C) is our answer.

If it is unclear if the given expressions do in fact give distinct linear equations, then basic simplification can help:
We'll rewrite \(8=2^3\), \(16=2^4\) and \(27=3^3\). This gives the following:
(1) \(2^{3(x + 2)} = 2^{4(4 + y)}\)
(2) \(3^{(x − y)} =3^{3(y + 6)}\)
Then (1) gvies 3(x+2)=4(4+y) and (2) gives (x-y)=3(y+6).
Now it should be clear that these are 2 distinct linear equations with two variables which is sufficient.
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Re: What is the value of x? (1) 8^(x + 2) = 16^(4 + y) (2) 3^(x − y) =27  [#permalink]

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New post 05 Apr 2018, 05:17
Bunuel wrote:
What is the value of x?

(1) \(8^{(x + 2)} = 16^{(4 + y)}\)

(2) \(3^{(x − y)} =27^{(y + 6)}\)


Statement 1: \(8^{(x + 2)} = 16^{(4 + y)}\)

i.e. \(2^{3(x + 2)} = 2^{4(4 + y)}\)

i.e. \({3(x + 2)} = {4(4 + y)}\)

i.e. \(3x + 6 = 16 + 4y\)

i.e. \(3x-4y = 10\)

NOT SUFFICIENT

Statement 2: \(3^{(x − y)} =27^{(y + 6)}\)

i.e. \(3^{(x − y)} =3^{3(y + 6)}\)

i.e. \(x-y = 3y+18\)

i.e. \(x-4y = 18\)

NOT SUFFICIENT

Combining the two statements

WE HAVE TWO EQUATIONS IN TWO VARIABLES AND OSLVING GTHEM WILL GIVE US UNIQUE VALUES OF X AND Y HENCE

SUFFICIENT

ANSWER: OPTION C
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Re: What is the value of x? (1) 8^(x + 2) = 16^(4 + y) (2) 3^(x − y) =27 &nbs [#permalink] 05 Apr 2018, 05:17
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