Solve the equation using elimination method: $5x + 6y = 9$ and $x - y = 4$
A) $( - 3, - 1)$
B) $(3, - 1)$
C) $( - 3,2)$
D) $(3,1)$

Hint: In the above question we will find the solutions of the equation using elimination method. This method eliminates one variable by adding or subtracting the equations and, allowing us to solve for the other. Since there are two variables in a two-equation scheme, replacing one makes solving for the other an ease.

Complete step by step solution:
Now, the equations are $5x + 6y = 9$ and $x - y = 4$
We will first multiply the second equation with $6$, so that the coefficient of $y$ variable is the same and opposite in both the equations.
Hence, the new equation is $6x - 6y = 24$
Now, adding this equation with the first equation we get,
\[
  5x + {{6y}} = 9 \\
  6x - {{6y}} = 24 \\
  \overline {11x + 0y = 33} \\
  11x = 33 \\
  x = \dfrac{{33}}{{11}} \\
  x = 3 \\
 \]
Hence, we eliminated the $y$ variable using the elimination process and calculated the value of $x$ variable. Now , we will substitute this value to the main equation to calculate the value of variable $y$ .
$
   \Rightarrow x - y = 4 \\
   \Rightarrow 3 - y = 4 \\
   \Rightarrow 3 - 4 = y \\
   \Rightarrow - 1 = y \\
  y = - 1 \\
 $
Hence, the values are $x = 3,y = - 1$. That is $(3, - 1)$. Therefore, the correct option is (B) (3,-1).

Note:
Elimination method is the most easiest method compared to all other processes for finding solutions of equations. They have the least number of steps and hence are quicker. We just have to ensure that we do not make mistakes in substituting the signs while adding or subtracting.