Simultaneous Equation With Two Unknowns

Theorem
A pair of simultaneous linear equations of the form:

where $a e \ne b d$, has as its only solution:

Proof 1
The solution for $x$ can be found similarly.

When $a e = b d$ we have that $a e - b d = 0$ and hence no solution exists.

Proof 2
This is an example of Solution to Simultaneous Linear Equations and can be solved using the technique of matrices.