Linear Systems with Addition or Subtraction

Key Questions

• The elimination method reduces the problem to solving a one variable equation.

For example, look at the following system of two variables:

$2 x + 3 y = 1$
$- 2 x + y = 7$

It is relatively difficult to determine the values of $x$ and $y$ without manipulating the equations. If one adds the two equations together, the $x$s cancel out; the $x$ is eliminated from the problem. Hence it is called the "elimination method."

One ends up with:

$4 y = 8$

From there, it is trivial to find $y$, and one can simply plug the value of $y$ back into either equation to find $x$.

• For me, addition method is safer (not necessarily easier) to use.
In subtraction method, switching the signs of each term of the second equation is prone to error.

• Let us solve the linear system below.

$\left\{\begin{matrix}3 x + y = 7 \\ x + 2 y = - 1\end{matrix}\right.$

The goal is to eliminate one of the variables by addition or subtraction so that the value of the other variable can be found.

By multiply the first equation by 2,

$\implies \left\{\begin{matrix}6 x + 2 y = 14 \\ x + 2 y = - 1\end{matrix}\right.$

by subtracting the second equation from the first equation,

$\implies 5 x = 15 \implies x = 3$

By plugging $x = 3$ in the first equation of the original system,

$\implies 3 \left(3\right) + y = 7 \implies 9 + y = 7 \implies y = - 2$

Hence, the solution is $\left(x , y\right) = \left(3 , - 2\right)$.

I hope that this was helpful.