Proof of Cramer's rule for two variables

Back to Cramer's rule

The standard equations are:
a₁₁x₁ + a₁₂x₂ = b₁
a₂₁x₁ + a₂₂x₂ = b₂

Solve the standard equations through linear combinations:
The first step:
a₂₂ (a₁₁x₁ + a₁₂x₂ = b₁)
-a₁₂ (a₂₁x₁ + a₂₂x₂ = b₂)

This equals:
a₁₁a₂₂x₁ + a₁₂a₂₂x₂ = b₁a₂₂
-a₁₂a₂₁x₁ + -a₁₂a₂₂x₂ = b₂a₁₂

Next we add the two equations:
(a₁₁a₂₂x₁ + a₁₂a₂₂x₂ = b₁a₂₂)
+
(-a₁₂a₂₁x₁ + -a₁₂a₂₂x₂ = b₂a₁₂)
=
(a₁₁a₂₂ - a₁₂a₂₁) x₁ = b₁a₂₂ - b₂a₁₂

When a₁₁a₂₂ - a₁₂a₂₁ does not equal 0:
x₁ = b₁a₂₂ - b₂a₁₂ / a₁₁a₂₂ - a₁₂a₂₁
and x₂ = b₂a₁₁ - b₁a₂₁ / a₁₁a₂₂ - a₁₂a₂₁

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