System of Simultaneous Equations may have No Solution

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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 empty.


Proof

Consider this system of simultaneous linear equations:

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

From its evaluation it is seen to have no solutions.

Hence the result.

$\blacksquare$


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