System of Simultaneous Equations may have Multiple Solutions
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Theorem
Let $S$ be a system of simultaneous equations.
Then it is possible that $S$ may have a solution set which is a singleton.
Proof
Consider this system of simultaneous linear equations:
\(\text {(1)}: \quad\) | \(\ds x_1 - 2 x_2 + x_3\) | \(=\) | \(\ds 1\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds 2 x_1 - x_2 + x_3\) | \(=\) | \(\ds 2\) |
From its evaluation it has the following solutions:
\(\ds x_1\) | \(=\) | \(\ds 1 - \dfrac t 3\) | ||||||||||||
\(\ds x_2\) | \(=\) | \(\ds \dfrac t 3\) | ||||||||||||
\(\ds x_3\) | \(=\) | \(\ds t\) |
where $t$ is any number.
Hence the are as many solutions as the cardinality of the domain of $t$.
$\blacksquare$
Sources
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.1$ Introduction