How do you solve the system using the elimination method for $3x-4y=18$ and $x+3y=-7$?

Hint: In this question we have been given two linear equations which are called simultaneous equations. We will solve the set of equations by using elimination methods. We will first multiply the second equation by $3$ to get the similar coefficient of $x$and then do subtraction and simplify to get the value of $x$. We will then substitute the value of $x$ in any equation to get the value of $y$.

Complete step-by-step answer:
We have the given equations as:
Equation $\left( 1 \right)$ can be written as: $3x-4y=18\to \left( 1 \right)$
Equation $\left( 2 \right)$ can be written as: $x+3y=-7\to \left( 2 \right)$
Now since we have to use elimination methods, we will try to get the coefficient of one variable the same value. On multiplying equation $\left( 2 \right)$ with $3$, we get:
$3x+9y=-21\to \left( 3 \right)$
Now since the equations are linear and the coefficient of $x$ is the same for both the equations, we don’t have to multiply or divide the equations, we will simplify both the equations just by simply subtracting them.
On subtracting equation $\left( 1 \right)$ from equation $\left( 3 \right)$ we get:
$\Rightarrow 13y=-39$
On transferring the term $13$ from the left-hand side to the right-hand side, we get:
$\Rightarrow y=\dfrac{-39}{13}$
On simplifying, we get:
$\Rightarrow y=-3$, which is the value of $y$.
Now to get the value of $x$ we will substitute the value of $y=-3$ in equation $\left( 2 \right)$.
On substituting, we get:
$\Rightarrow x+3\left( -3 \right)=-7$
On simplifying the bracket, we get:
$\Rightarrow x-9=-7$
On transferring the term $-9$ from the left-hand side to the right-hand side, we get:
$\Rightarrow x=9-7$
On simplifying, we get:
$\Rightarrow x=2$
Therefore, the values of $x$ and $y$ are $2$ and $-3$ respectively, which is the required solution.

Note: It is to be remembered that in any given equation multiplying or dividing the equation by a specific constant doesn’t change the value of the equation. In the given question we had two variables which are $x$ and $y$ , therefore they can be solved by using elimination, where there are more than three variables, and the matrix is used to solve them.