Question

Suppose that $x$ and $y$ are real numbers that satisfy the system of equations
$2^x-2^y=1$,
$4^x-4^y=\frac{5}{3}$
The value of $(x-y)$ is

• A. $3$
• B. $2$
• C. $1$
• D. $4$

Answer 1

The correct option is B $2$

$\frac{5}{3}=4^x-4^y=(2^x-2^y)(2^x+2^y)=1.(2^x+2^y)$
Therefore, $2^x=\frac{(2^x+2^y)+(2^x-2^y)}{2}=\frac{\frac{5}{3}+1}{2}=\frac{4}{3}$
And $2^y=\frac{(2^x+2^y)-(2^x-2^y)}{2}=\frac{\frac{5}{3}-1}{2}=\frac{1}{3}$
Which implies $2^{x-y}=\frac{2^x}{2^y}=\frac{4/3}{1/3}=4$
and thus $x-y=2.$