Simultaneous Linear Equations/Examples/Arbitrary System 1

Example of Simultaneous Linear Equations
The system of simultaneous linear equations:

has as its solution set:

Proof
Subtract $2 \times$ equation $(1)$ from equation $(2)$.

Subtract $4 \times$ equation $(1)$ from equation $(3)$.

This gives us:

Divide equation $(2')$ by $3$ to get $(2'')$.

Add $2 \times$ equation $(2'')$ to equation $(1)$.

Subtract $9 \times$ equation $(2'')$ from equation $(3')$.

This gives us:

From equation $(3'')$ we have directly that $x_3 = \dfrac 3 2$.

Substituting for $x_3$ in equation $(1')$ and equation $(2'')$ gives the single solution: