Question

Solve for $ x $ and $ y $ :
 $ {3^x} = {9.3^y},{8.2^y} = {4^x} $

Answer 1

Hint: If you have ​one​ equation with two variables, you'll probably be asked to solve for just one of those variables.In that case you follow much the same procedure as you'd use for any algebraic equation with one variable. Here we use laws of exponents to get the equations and then solve them to find the value of x and y.

Complete step-by-step answer:
First we can write it as,
 $ \Rightarrow $ $ {3^x} = ({3^2}){.3^y},({2^3}){.2^y} = {4^x} $
 $ \Rightarrow {3^x} = {3^{2 + y}} $ and,
 $ \Rightarrow {2^{3 + y}} = {2^{2x}} $
On comparing powers of the both the equations,
We get,
 $
   \Rightarrow x = 2 + y \\
   \Rightarrow 2x = 3 + y \;
  $
Solve these two equations,
 $
   \Rightarrow 2\left( {y + 2} \right) = 3 + y \\
   \Rightarrow 2y + 4 = 3 + y \\
   \Rightarrow y = - 1. \;
  $
On substituting we get,
 $ \Rightarrow x = 1 $
Therefore, $ x = 1,y = - 1. $
So, the correct answer is “$ x = 1,y = - 1. $”.

Note: $ \Rightarrow $ If you're given a system of ​two​ equations that have the same two variables in them, this usually means the equations are related and you can use a technique called substitution to find values for both variables.
 $ \Rightarrow $ Doing the same thing to both sides of the equation keeps the equation balanced.
And thus, the balanced equation is easily solved.