# Question #1553a

Jun 6, 2017

Eliminating one variable allows you to solve for the other.

#### Explanation:

You cannot solve an equation that has $2$ variables because there are infinitely many solutions.

As soon as you have 2 equations, there is a specific solution.

However, you need to get rid of one of the variables temporarily so that you can find a value for the one that is left.

This is done by making sure that the variable that is to be eliminated has the same numerical coefficients.

Case 1: If they have the SAME sign, subtract the two equations:

Consider the equations below:

$5 x \textcolor{b l u e}{+ 5 y} = 30$
$3 x \textcolor{b l u e}{+ 5 y} = 22$

Subtracting the two equations will lead to

$2 x \textcolor{b l u e}{+ 0 y} = 8 \text{ } \leftarrow$ the $y$-terms have been eliminated

The difference between the two equations represents the difference in just the $x$ terms.

Case 2: If they have different signs, ADD the two equations:

Consider the equations below: the $x$-terms are additive inverses

$\textcolor{red}{+ 4 x} + 3 y = 29$
$\textcolor{red}{- 4 x} + 5 y = - 5$

$\textcolor{red}{0 x} + 8 y = 24 \text{ } \leftarrow$ the $x$-terms have been eliminated
The sum of the equations gives us the sum of just the $y$-terms