= 2 Equations, 2 Unknowns =

== Introduction ==

Consider the below system of equations:

{{{
2x - y = 0
-x + 2y = 3
}}}

This can be envisioned in three ways.



== Row Picture ==

The equations can be plotted together, and all intersections of the lines are solutions to the system.

{{attachment:row-picture.png||width=200}}



== Column Picture ==

The system is equivalent to the following '''linear combination of columns'''.

{{{
 ┌  ┐    ┌  ┐   ┌  ┐
 │ 2│    │-1│   │ 0│
x│-1│ + y│ 2│ = │ 3│
 └  ┘    └  ┘   └  ┘ 
}}}

Column 1 is represented as a vector to (2,-1); column 2 as (-1,2).

{{attachment:column-picture.png||width=200}}

Any combination of these vectors that leads to (0,3) is a solution to the system. For this system, that solution is 1 of column 1 and 2 of column 2.

{{attachment:column-picture-solution.png||width=200}}



== Matrix Picture ==

{{{
┌     ┐   ┌  ┐   ┌  ┐
│ 2 -1│   │ x│   │ 0│
│-1  2│ + │ y│ = │ 3│
└     ┘   └  ┘   └  ┘ 
}}}



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