System of Simultaneous Equations may have Multiple Solutions

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Let $S$ be a system of simultaneous equations.

Then it is possible that $S$ may have a solution set which is a singleton.


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 $t$.