If 6^x - 3*6^(x-4) = 1293*16^y, what is the value of x, in terms of y?

Question

If  6^x-3*6^{x-4}=1293*36^y , what is the value of x, in terms of y?

(A) y
(B) y+4
(C) y-4
(D) 2y+4
(E) 2y-4

chetan2u · Answer

The terms in the expression tell you to separate out terms with base 6.

6^x-3*6^{x-4}=1293*36^y

6^x-3*\frac{6^x}{6^4}=1293*36^y

6^{x+4}-3*6^{x}=1293*(6^2)^y*6^4

6^x(6^4-3)=1293*6^{2y+4}

6^x*1293=1293*6^{2y+4}

6^x=6^{2y+4}

x=2y+4

D

ppate · Answer

Could we instead set X = 4 making the RHS

6^x-3*6^{x-4}=1293*36^y

6^4-3*6^{4-4}=1293*36^y

1296-3 = 1293*36^y

1293 = 1293*36^y

1 = 36^y

y = 0

When y = 0, x must equal 4.

Going through the answer choices, plug in when Y = 0
x | y
4 = 0 X
4 = 8 X
4 = -4 X
 4 = 4 Good  
4 = -4 X

chetan2u · Answer

Even B gives you 4=4, as 2y+4=4

Your method is correct but you can land up with two options in certain cases. 
In these cases, you may have to take a second value of x to eliminate one of the options.

Yes, had you taken x=2, you would have got just one answer.

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If 6^x - 36^(x-4) = 129316^y, what is the value of x, in terms of y?

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