Question

If $3^{x+y}=81and{81}^{x-y}=3$, then the value of x and y are

• A. $\frac{17}{8},\frac{9}{8}$
• B. $\frac{17}{8},\frac{15}{8}$
• C. $\frac{17}{8},\frac{11}{8}$
• D. $\frac{15}{8},\frac{11}{8}$

Answer 1

The correct option is B $\frac{17}{8},\frac{15}{8}$

$3^{x+y}=81\Rightarrow 3^{x+y}=3^4=x+y=4$ ......... $\left(i\right)$
${81}^{x-y}=3\Rightarrow (3^4{)}^{x-y}=3^1$
$\Rightarrow 4x-4y=1$ .......... $\left(ii\right)$
Multiply eqn $\left(i\right)$ by $4$ and then add with eqn $\left(ii\right)$, gives
$4x+4y+4x-4y=16+1$
$\Rightarrow 8x=17\Rightarrow x=\frac{17}{8}$
Putting $x=\frac{17}{8}$ in $\left(i\right)$ we get

$\frac{17}{8}+y=4$
$\Rightarrow y=4-\frac{17}{8}=\frac{15}{8}$

$\therefore x=\frac{17}{8},y=\frac{15}{8}$