Question

Solve the following equations:

$\frac{7}{3^x}-\frac{6}{2^y}=15$; $\frac{8}{3^x}=\frac{9}{2^y}$

• A. $(-2,-3)$
• B. $(-1,2)$
• C. $(-4,5)$
• D. $(0,9)$

Answer 1

The correct option is A $(-2,-3)$

Given equations are,

$\frac{7}{3^x}-\frac{6}{2^y}=15$ ....(1)

and $\frac{8}{3^x}=\frac{9}{2^y}$ ....(2)

Therefore, $3^x=\frac{8\times 2^y}{9}$ ....(3)

Put this value in equation (1), we get

$\frac{63}{8\times 2^y}-\frac{6}{2^y}=15$

$\Rightarrow \frac{1}{2^y}\left(\frac{63-48}{8}\right)=15$

$\Rightarrow 1=8\times 2^y$

$\Rightarrow 2^y=2^{-3}$

$\Rightarrow y=-3$

Put this value in equation (3),

$3^x=\frac{8\times 2^{-3}}{9}$

$\Rightarrow 3^x=3^{-2}$

$\Rightarrow x=-2$

Therefore, solution is $(x,y)=(-2,-3)$.