Editing Elementary algebra (section)

=== Other types of systems of linear equations ===

==== 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.

==== Undetermined systems ====

There are also systems which have infinitely many solutions, in contrast to a system with a unique solution (meaning, a unique pair of values for {{mvar|x}} and {{mvar|y}}) For example:

: <math>\begin{cases}\begin{align}4x + 2y & = 12 \\
-2x - y & = -6 \end{align}\end{cases}</math>

Isolating {{mvar|y}} in the second equation:

: <math>y = -2x + 6 </math>

And using this value in the first equation in the system:

: <math>\begin{align}4x + 2(-2x + 6) = 12 \\
4x - 4x + 12 = 12 \\
12 = 12 \end{align}</math>

The equality is true, but it does not provide a value for {{mvar|x}}. Indeed, one can easily verify (by just filling in some values of {{mvar|x}}) that for any {{mvar|x}} there is a solution as long as <math>y = -2x + 6</math>. There is an infinite number of solutions for this system.

==== Over- and underdetermined systems ====

Systems with more variables than the number of linear equations are called [[underdetermined system|underdetermined]]. Such a system, if it has any solutions, does not have a unique one but rather an infinitude of them. An example of such a system is

: <math>\begin{cases}\begin{align}x + 2y & = 10\\
y - z  & = 2 .\end{align}\end{cases}</math>

When trying to solve it, one is led to express some variables as functions of the other ones if any solutions exist, but cannot express ''all'' solutions [[Number|numerically]] because there are an infinite number of them if there are any.

A system with a higher number of equations than variables is called [[overdetermined system|overdetermined]]. If an overdetermined system has any solutions, necessarily some equations are [[linear combination]]s of the others.