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.
$\blacksquare$
Sources
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.1$ Introduction