Simultaneous Linear Equations/Examples/Arbitrary System 3

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

has as its solution set:

where $t$ is any number.

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

This gives us:

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

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

This gives us:

That is:

Whatever value is assigned to $x_3$, the system of simultaneous linear equations will be fulfilled.

Thus we set $x_3$ to any arbitrary $t$ and the result follows.