Question

If ${\left( 3 \right)^{x + y}} = 81$ and ${\left( {81} \right)^{x - y}} = 3$, then the values of $x$ and $y$ are
A. $\dfrac{{17}}{8},\dfrac{9}{8}$
B. $\dfrac{{17}}{8},\dfrac{{11}}{8}$
C. $\dfrac{{17}}{8},\dfrac{{15}}{8}$
D. $\dfrac{{11}}{8},\dfrac{{15}}{8}$

Answer 1

Hint: To solve such questions start by simplifying each given term. First, consider one term and then equate the left-hand side and right-hand side of the term. Similarly consider the other term and equate the LHS and RHS. Computing these two equations the values of $x$ and $y$ are obtained.

Complete step-by-step solution:
Given ${\left( 3 \right)^{x + y}} = 81$ and ${\left( {81} \right)^{x - y}} = 3$
It is asked to find the value of $x$ and $y$.
First, consider the term ${\left( 3 \right)^{x + y}} = 81$ .
The RHS term $\;81$ can be written as $3$ raise to the power of $4$ .
That is,
$\Rightarrow$$81 = {3^4}$
Equating this to the LHS, that is,
$\Rightarrow$${\left( 3 \right)^{x + y}} = {\left( 3 \right)^4}$
Comparing the powers, we get,
$\Rightarrow$$x + y = 4$
$\Rightarrow$$x = 4 - y……………....(1)$
Next, consider the term ${\left( {81} \right)^{x - y}} = 3$
We know that $\;81$can be written as $3$ raise to the power of $4$ .
Therefore,
$\Rightarrow$${\left( 3 \right)^{4\left( {x - y} \right)}} = 3$
Comparing the powers, we get,
$\Rightarrow$$4\left( {x - y} \right) = 1$
$\Rightarrow$$4x - 4y = 1……………….....(2)$
Substituting equation $\left( 1 \right)$in the equation $\left( 2 \right)$ , we get
$\Rightarrow$$4\left( {4 - y} \right) - 4y = 1$
$\Rightarrow$$16 - 4y - 4y = 1$
Simplifying further, we get,
$\Rightarrow$$- 8y = 1 - 16$
$\Rightarrow$$- 8y = - 15$
Cancel out the minus sign on both sides, that is,
$\Rightarrow$$8y = 15$
Dividing through by $8$ , we get,
$\Rightarrow$$\dfrac{{8y}}{8} = \dfrac{{15}}{8}$
Canceling out the common term we get,
$\Rightarrow$$y = \dfrac{{15}}{8}$
Substitute the value of $y$ in equation $\left( 1 \right)$ , we get,
$\Rightarrow$ $x = 4 - \dfrac{{15}}{8}$
$\Rightarrow$$x = \dfrac{{4 \times 8 - 15}}{8}$
Simplifying further, we get,
$\Rightarrow$$x = \dfrac{{32 - 15}}{8}$
$\Rightarrow$$x = \dfrac{{17}}{8}$
Therefore the value of $x$ and $y$ are $x = \dfrac{{17}}{8}$ and $y = \dfrac{{15}}{8}$.

Option C is the correct answer.

Note: Two linear equations in the same two variables are known as a pair of linear equations in two variables. A pair of linear equations in two variables can be represented, and solved, by the graphical method and the algebraic method.
After computing the values of $x$ and $y$ we can find whether the obtained value is correct or not by substituting these values in the given equation. When the given equations are not linear, then we can make changes in them so that they become linear equations.