Question

If $3x - 2y = 24$ and $2x - 3y = 18$ then find the value of $x + y$ and $x - y$

Answer 1

Hint:
It is given in the question that if $3x - 2y = 24$ and $2x - 3y = 18$.
Then, we have to find the value of $x + y$ and $x - y$.
Using the method of elimination on the given pair of equations, find the value of x and y.
Thus, find the required value of $x + y$ and $x - y$.

Complete step by step solution:
It is given in the question that if $3x - 2y = 24$ and $2x - 3y = 18$ .
Then, we have to find the value of $x + y$ and $x - y$ .
 $3x - 2y = 24$ (I)
 $2x - 3y = 18$ (II)
Now, we are going to find the values of x and y by using the elimination method.
First, we will multiply equation (I) with 2, we get,
 $2\left( {3x - 2y = 24} \right)$
 $6x - 4y = 48$ (III)
Now, we will multiply equation (II) with -3, we get,
 $ - 3\left( {2x - 3y = 18} \right)$
 $ - 6x + 9y = - 54$ (IV)
Now, by the method of elimination we will find the values of x and y by eliminating equation (III) and (IV).
 $
   - 6x + 9y = - 54 \\
   + 6x - 4y = 48 \\
  \_\_\_\_\_\_\_\_\_\_\_\_ \\
  0 + 5y = - 6 \\
 $
 $\Rightarrow 5y = - 6$
 $\Rightarrow y = - \dfrac{6}{5}$
Now, put the value of ‘y’ in equation (I), we get
 $\Rightarrow 3x - 2y = 24$
 $\Rightarrow 3x - 2\left( {\dfrac{{ - 6}}{5}} \right) = 24$
 $\Rightarrow 3x + \dfrac{{12}}{5} = 24$
 $\Rightarrow 3x = 24 - \dfrac{{12}}{5}$
 $\Rightarrow 3x = \dfrac{{120 - 12}}{5}$
 $\Rightarrow 3x = \dfrac{{108}}{5}$
 $\Rightarrow x = \dfrac{{108}}{{5 \times 3}}$
 $\Rightarrow x = \dfrac{{108}}{{15}}$
 $\Rightarrow x = \dfrac{{36}}{5}$
Therefore, the value of $x = \dfrac{{36}}{5}$ and $y = - \dfrac{6}{5}$
Now,
 $x + y = \dfrac{{36}}{5} + \left( { - \dfrac{6}{5}} \right)$
 $\Rightarrow x + y = \dfrac{{36}}{5} - \dfrac{6}{5}$
 $\Rightarrow x + y = \dfrac{{36 - 6}}{5}$
 $\Rightarrow x + y = \dfrac{{30}}{5}$
 $\Rightarrow x + y = 6$
Similarly,
 $\Rightarrow x - y = \dfrac{{36}}{5} - \left( { - \dfrac{6}{5}} \right)$
 $\Rightarrow x - y = \dfrac{{36}}{5} + \dfrac{6}{5}$
 $\Rightarrow x - y = \dfrac{{36 + 6}}{5}$
 $\Rightarrow x - y = \dfrac{{42}}{5}$

Hence, the value of $x + y = 6$ and $x - y = \dfrac{{42}}{5}$.

Note:
Elimination method: In an elimination method we either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites, we add the equations to eliminate a variable and when the coefficients of one variable are equal, we subtract the equations to eliminate a variable.