How do you solve $5x - 3y = - 14\;{\text{and}}\;x - 3y = 2$ using elimination?

Hint: To solve the equations elimination method, the first step is to multiply both or one equation with a number so that the coefficient one of the variables (x or y) of both equations becomes equal. After that, add or subtract both equations to cancel the variable with the same coefficient and then solve the rest equation for the other variable and finally substitute the value of the other variable in any of the two equations to get the solution.

Complete step by step solution:
In order to solve the given equations $5x - 3y = - 14\;{\text{and}}\;x - 3y = 2$ using elimination method, we need to coefficient of one of the variable similar, but we can see that in the given equations $5x - 3y = - 14\;{\text{and}}\;x - 3y = 2$, coefficient of $y$ is already similar.
So now subtracting the second equation from the first one in order to eliminate $y$ variable,
\[
  5x - 3y = - 14\; \\
  \underline { - x + 3y = - 2} \\
  4x + 0 = - 16 \\
 \]
So after elimination, we get \[4x = - 16\], now solving this equation for value of $x$
\[ \Rightarrow 4x = - 16\]
Dividing both sides with $4$ we will get
\[
   \Rightarrow \dfrac{{4x}}{4} = \dfrac{{ - 16}}{4} \\
   \Rightarrow x = - 4 \\
 \]
Now, substituting \[x = - 4\] in the second equation to get the value of $y$
\[ \Rightarrow - 4 - 3y = 2\]
Adding $4$ both sides,
\[
   \Rightarrow - 4 - 3y + 4 = 2 + 4 \\
   \Rightarrow - 3y = 6 \\
 \]
Now, dividing both sides with $3$ and then multiplying $ - 1$ we will get
\[
   \Rightarrow \dfrac{{ - 3y}}{3} = \dfrac{6}{3} \\
   \Rightarrow - y = 2 \\
   \Rightarrow - 1 \times ( - y) = - 1 \times 2 \\
   \Rightarrow y = - 2 \\
 \]
Therefore $x = - 4\;{\text{and}}\;y = - 2$ is the required solution for the given equation.

Note: When after eliminating you are getting no variable, then check for the calculations and signs in the steps and if everything is ok even you are getting no variable after the elimination and if the statement after elimination is true then the set of equations are collinear and have infinite solutions and if the statement is not true then the set of equations are parallel and inconsistent having zero solutions.