Question

For \[2x + 3y = 7\] and $3x + 2y = 3$, what is the value of $x - y$ ?
A. $4$
B. $- 4$
C. $10$
D. $21$

Answer 1

Hint: We can evaluate the value of $x$ and $y$ by substitution method. Evaluate the value of $x$ in terms of $y$ from one equation and substitute in the other equation to find the value of $x$. Substitute the value of $x$ in any of the equations to evaluate the value of $y$. Substitute the values in the expression $x - y$

Complete step-by-step answer:
We are given two linear equations. Number the equations.
\[2x + 3y = 7....(1)\]
\[3x + 2y = 3....(2)\]
We have to find the value of $x - y$.
First, we evaluate the value of $y$.
Evaluate the value of $x$ in terms of $y$ from equation $(1)$
$2x = 7 - 3y$
Divide both sides by $2$.
$x = \dfrac{{7 - 3y}}{2}$
Substitute the value of $x$ in equation $(2)$
$3\left( {\dfrac{{7 - 3y}}{2}} \right) + 2y = 3$
Solve the equation and find the value of $y$.
$
  21 - 9y + 4y = 6 \\
   \Rightarrow - 5y = - 15 \\
   \Rightarrow y = 3 \\
$
We have evaluated the value of $y$.
Now, we evaluate the value of $x$.
To evaluate the value of $x$, substitute the value of $y$ in any of the equations.
Suppose we substitute the value of $y$ in equation $(1)$.
$2x + 3(3) = 7$
Solve the equation and find the value of $x$.
$
  2x + 9 = 7 \\
   \Rightarrow 2x = - 2 \\
   \Rightarrow x = - 1 \\
 $
Now, we evaluate the value of $x - y$
Substitute the values of $x$ and $y$in the expression $x - y$.
$
  x - y = - 1 - 3 \\
   \Rightarrow x - y = - 4 \\
 $
Therefore, the value of $x - y$ is $ - 4$.

So, the correct answer is “Option B”.

Note: We can solve this question by another method which is shown below:
In this question we can directly evaluate the value of $x - y$ without evaluating the values of $x$ and $y$.
We are given two linear equations.
\[2x + 3y = 7....(1)\]
\[3x + 2y = 3....(2)\]
Subtract the equation $(1)$ from the equation $(2)$
$3x + 2y - (2x + 3y) = 3 - 7$
Solve the equation and you will get the value of $x - y$.
$
  3x + 2y - 2x - 3y = - 4 \\
   \Rightarrow x - y = - 4 \\
 $
Therefore, the value of $x - y$ is $ - 4$.
Hence, option (B) is correct.