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If $3x - 2y = 24$ and $2x - 3y = 18$ then find $x + y$ and $x - y$.

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Last updated date: 13th Jun 2024
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Answer
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Hint: In this problem, we have two linear equations in two variables $x$ and $y$. First we will solve these two equations by a simple elimination method. Therefore, we will get values of $x$ and $y$. By adding and subtracting values of $x$ and $y$, we will get required values.

Complete step-by-step answer:
In this problem, two linear equations in two variables $x$ and $y$ are given. Let us write the given equations.
$3x - 2y = 24 \cdots \cdots \left( 1 \right)$
$2x - 3y = 18 \cdots \cdots \left( 2 \right)$
Now we are going to find values of $x$ and $y$ by using a simple elimination method. In this method, first we will multiply each equation by a suitable number such that coefficients of variable $x$ or $y$ become equal in both equations. For this, let us multiply both sides of the equation $\left( 1 \right)$ by number $2$. Therefore, we get $6x - 4y = 48 \cdots \cdots \left( 3 \right)$. Now we will multiply on both sides of equation $\left( 2 \right)$ by number $3$. Therefore, we get $6x - 9y = 54 \cdots \cdots \left( 4 \right)$
Now we can see that in equations $\left( 3 \right)$ and $\left( 4 \right)$ coefficient of $x$ is equal. Now to find the value of $y$, we need to eliminate $x$. For that, we will subtract the equation $\left( 3 \right)$ from the equation $\left( 4 \right)$. Therefore, we get
$
  \left( {6x - 9y} \right) - \left( {6x - 4y} \right) = 54 - 48 \\
   \Rightarrow 6x - 9y - 6x + 4y = 6 \\
   \Rightarrow - 9y + 4y = 6 \\
   \Rightarrow - 5y = 6 \\
   \Rightarrow y = - \dfrac{6}{5} \\
 $
Now to find the value of $x$, we will substitute the value of $y$ in either equation $\left( 1 \right)$ or equation $\left( 2 \right)$. Let us substitute the value of $y$ in the equation $\left( 1 \right)$ and simplify it. Therefore, we get
$
  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{{36}}{5} \\
 $
Now we have values of $x$ and $y$. That is, $x = \dfrac{{36}}{5}$ and $y = - \dfrac{6}{5}$. Now we will add these two values to find $x + y$ and we will subtract these two values to find $x - y$. Therefore, we get
$
  x + y = \dfrac{{36}}{5} + \left( { - \dfrac{6}{5}} \right) \\
   \Rightarrow x + y = \dfrac{{36 - 6}}{5} \\
   \Rightarrow x + y = \dfrac{{30}}{5} \\
   \Rightarrow x + y = 6 \\
 $
$
  x - y = \dfrac{{36}}{5} - \left( { - \dfrac{6}{5}} \right) \\
   \Rightarrow x - y = \dfrac{{36 + 6}}{5} \\
   \Rightarrow x - y = \dfrac{{42}}{5} \\
 $
Therefore, if $3x - 2y = 24$ and $2x - 3y = 18$ then $x + y = 6$ and $x - y = \dfrac{{42}}{5}$.


Note: Let us try to solve the given problem by different method.There is also a shortcut method which exists to solve the linear equations. Also note that we can solve two linear equations by using matrix inversion method.