Editing System of linear equations (section)

==Elementary examples==

===Trivial example===
The system of one equation in one unknown
: <math>2x = 4</math>
has the solution 
: <math>x = 2.</math>

However, most interesting linear systems have at least two equations.

===Simple nontrivial example===
The simplest kind of nontrivial linear system involves two equations and two variables:

: <math>\begin{alignat}{5}
2x &&\; + \;&& 3y &&\; = \;&& 6 & \\
4x &&\; + \;&& 9y &&\; = \;&& 15&.
\end{alignat}</math>

One method for solving such a system is as follows. First, solve the top equation for <math>x</math> in terms of <math>y</math>:

: <math>x = 3 - \frac{3}{2}y.</math>

Now [[substitution (algebra)|substitute]] this expression for ''x'' into the bottom equation:

: <math>4\left( 3 - \frac{3}{2}y \right) + 9y = 15.</math>

This results in a single equation involving only the variable <math>y</math>. Solving gives <math>y = 1</math>, and substituting this back into the equation for <math>x</math> yields <math>x = \frac{3}{2}</math>. This method generalizes to systems with additional variables (see "elimination of variables" below, or the article on [[elementary algebra]].)