Editing Elementary algebra (section)

==== Inconsistent systems ====
[[File:Parallel Lines.svg|thumb|right|The equations <math>3x + 2y = 6</math> and <math>3x + 2y = 12</math> are parallel and cannot intersect, and is unsolvable.]]
[[File:Quadratic-linear-equations.svg|thumb|right|Plot of a quadratic equation (red) and a linear equation (blue) that do not intersect, and consequently for which there is no common solution.]]

In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called [[inconsistent system|inconsistent]]. An obvious example is

: <math>\begin{cases}\begin{align} x + y &= 1 \\
0x + 0y &= 2\,. \end{align} \end{cases}</math>

As 0≠2, the second equation in the system has no solution. Therefore, the system has no solution.
However, not all inconsistent systems are recognized at first sight. As an example, consider the system 
: <math>\begin{cases}\begin{align}4x + 2y &= 12 \\
-2x - y &= -4\,. \end{align}\end{cases}</math>

Multiplying by 2 both sides of the second equation, and adding it to the first one results in
: <math>0x+0y = 4 \,,</math>
which clearly has no solution.