Question

Solve the following system of equations.
${3^x} \cdot {2^y} = 576$ , ${\log _{\sqrt 2 }}\left( {y - x} \right) = 4$

Answer 1

Hint : In this question, two equations are given and we have to solve them and find the value of $x{\rm{ and }}y$ variables. We first simplify one equation and find a relation between both the variables and then using that relationship, we can calculate the value of the variables.

Complete step-by-step answer :
The system of equations-
The first equation is,
${3^x} \cdot {2^y} = 576$
And, the second equation is,
${\log _{\sqrt 2 }}\left( {y - x} \right) = 4$
Now considering the second equation and solving it step by step, we get,
${\log _{\sqrt 2 }}\left( {y - x} \right) = 4$
In order to remove the logarithm, we have to raise both sides of the equation to the same value of exponent as the base of the log value. So, removing the ${\log _{\sqrt 2 }}$ by raising both sides to $\sqrt 2$ we get,
$$ $\begin{array}{c} \left( {y - x} \right) = {\left( {\sqrt 2 } \right)^4}\\\ \left( {y - x} \right) = 4\\\ y = 4 + x \end{array}$
So, now we have a relation between the variables $x{\rm{ and }}y$ given by
$y = 4 + x$
Now substituting this value of $y$ in the second equation and solving it for the values of $x{\rm{ and }}y$ -
${3^x} \cdot {2^{4 + x}} = 576$
We can write this as-
${3^x} \cdot {2^4} \cdot {2^x} = 576$
We know that ${a^x} \cdot {b^x} = {\left( {ab} \right)^x}$ and ${2^4} = 16$ , substituting this in the equation we get,
$\begin{array}{c} 16 \cdot {6^x} = 576\\\ {6^x} = \dfrac{{576}}{{16}}\\\ {6^x} = 36 \end{array}$
We can write $36$ as $36 = {6^2}$ , putting this in the equation we get,
${6^x} = {6^2}$
Comparing the exponents from both the sides we get,
$x = 2$
Substituting this value of $x$ in the first equation and solving it for $y$ we get,
$\begin{array}{c} {3^2} \cdot {2^y} = 576\\\ 9 \cdot {2^y} = 576\\\ {2^y} = \dfrac{{576}}{9}\\\ {2^y} = 64 \end{array}$
We can write $64$ as $64 = {2^6}$ , putting this in the equation we get,
${2^y} = {2^6}$
Comparing the exponents from both the sides we get,
$y = 6$
Therefore, The value of $x$ is $2$ and the value of $y$ is $6$ .

Note : If we consider the first equation ${3^x} \cdot {2^y} = 576$ and then from this equation, we try to find a relation between variables, $x{\rm{ and }}y$ the solution gets more complex and harder to solve. That is the reason we have considered the second equation and derived the relation between $x{\rm{ and }}y$ from it.