10th Class Mathematics Pair of Linear Equations in Two Variables Question Bank MCQs - Pair of Linear Equations in Two Variables

The pair of equations \[{{3}^{x+y}}=81,\,{{81}^{x-y}}=3\] has

A) no solution

B) unique solution

C) infinitely many solutions

D) \[x=2\frac{1}{8},\,y=1\frac{7}{8}\]

Correct Answer: D

Solution :

Given,

\[{{3}^{x+y}}=81\Rightarrow {{3}^{x+y}}={{3}^{4}}\Rightarrow x+y=4\] (i)

And \[{{81}^{x-y}}=3\Rightarrow {{3}^{4\left( x-y \right)}}={{3}^{1}}\Rightarrow 4\left( x-y \right)=1\]

\[\Rightarrow \,x-y=\frac{1}{4}\] (ii)

On adding Eqs. (i) and (ii), we get

\[2x=4+\frac{1}{4}=\frac{17}{4}\]

\[\Rightarrow \,\,\,x=\frac{17}{8}=2\frac{1}{8}\]

From Eq. (i), we get

\[y=\frac{15}{8}=1\frac{7}{8}\]