Editing Elementary algebra (section)

=== System of linear equations ===
{{Main|System of linear equations}}

There are different methods to solve a system of linear equations with two variables.

==== Elimination method ====
[[File:Intersecting Lines.svg|thumb|right|The solution set for the equations <math>x - y = -1</math> and <math>3x + y = 9</math> is the single point (2,&nbsp;3).]]

An example of solving a system of linear equations is by using the elimination method:

: <math>\begin{cases}4x + 2y&= 14 \\
2x - y&= 1.\end{cases} </math>

Multiplying the terms in the second equation by 2:

: <math>4x + 2y = 14 </math>
: <math>4x - 2y = 2. </math>

Adding the two equations together to get:

: <math>8x = 16 </math>

which simplifies to

: <math>x = 2. </math>

Since the fact that <math>x = 2</math> is known, it is then possible to deduce that <math>y = 3</math>  by either of the original two equations (by using ''2'' instead of {{mvar|x}} ) The full solution to this problem is then

: <math>\begin{cases} x = 2 \\ y = 3. \end{cases}</math>

This is not the only way to solve this specific system; {{mvar|y}} could have been resolved before {{mvar|x}}.

==== Substitution method ====

Another way of solving the same system of linear equations is by substitution.

: <math>\begin{cases}4x + 2y &= 14
\\ 2x - y &= 1.\end{cases} </math>

An equivalent for {{mvar|y}} can be deduced by using one of the two equations. Using the second equation:

: <math>2x - y = 1 </math>

Subtracting <math>2x</math> from each side of the equation:

: <math>\begin{align}2x - 2x - y & = 1 - 2x  \\
- y & = 1 - 2x
\end{align}</math>

and multiplying by −1:

: <math> y = 2x - 1. </math>

Using this {{mvar|y}} value in the first equation in the original system:

: <math>\begin{align}4x + 2(2x - 1) &= 14\\
4x + 4x - 2 &= 14 \\
8x - 2 &= 14 \end{align}</math>

Adding ''2'' on each side of the equation:

: <math>\begin{align}8x - 2 + 2 &= 14 + 2 \\
8x &= 16 \end{align}</math>

which simplifies to

: <math>x = 2 </math>

Using this value in one of the equations, the same solution as in the previous method is obtained.

: <math>\begin{cases} x = 2 \\ y = 3. \end{cases}</math>

This is not the only way to solve this specific system; in this case as well, {{mvar|y}} could have been solved before {{mvar|x}}.