Simultaneous Linear Equations/Examples/Arbitrary System 3

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Example of Simultaneous Linear Equations

The system of simultaneous linear equations:

\(\text {(1)}: \quad\) \(\ds x_1 - 2 x_2 + x_3\) \(=\) \(\ds 1\)
\(\text {(2)}: \quad\) \(\ds 2 x_1 - x_2 + x_3\) \(=\) \(\ds 2\)

has as its solution set:

\(\ds x_1\) \(=\) \(\ds 1 - \dfrac t 3\)
\(\ds x_2\) \(=\) \(\ds \dfrac t 3\)
\(\ds x_3\) \(=\) \(\ds t\)

where $t$ is any number.


Proof

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

This gives us:

\(\text {(1)}: \quad\) \(\ds x_1 - 2 x_2 + x_3\) \(=\) \(\ds 1\)
\(\text {(2')}: \quad\) \(\ds 3 x_2 - x_3\) \(=\) \(\ds 0\)


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

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

This gives us:

\(\text {(1')}: \quad\) \(\ds x_1 + \dfrac {x_3} 3\) \(=\) \(\ds 1\)
\(\text {(2'')}: \quad\) \(\ds x_2 - \dfrac {x_3} 3\) \(=\) \(\ds 0\)

That is:

\(\ds x_1 +\) \(=\) \(\ds 1 - \dfrac {x_3} 3\)
\(\ds x_2\) \(=\) \(\ds \dfrac {x_3} 3\)

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.

$\blacksquare$


Sources