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
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.1$ Introduction: Example $\text {(iii)}$