Simultaneous Linear Equations/Examples/Arbitrary System 2
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Example of Simultaneous Linear Equations
The system of simultaneous linear equations:
\(\text {(1)}: \quad\) | \(\ds x_1 + x_2\) | \(=\) | \(\ds 2\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds 2 x_1 + 2 x_2\) | \(=\) | \(\ds 3\) |
has no solutions.
Proof
Aiming for a contradiction, suppose $(1)$ and $(2)$ together have a solution.
Subtract $2 \times$ equation $(1)$ from equation $(2)$.
\(\text {(1)}: \quad\) | \(\ds x_1 - 2 x_2 + x_3\) | \(=\) | \(\ds 1\) | |||||||||||
\(\text {(2')}: \quad\) | \(\ds 0\) | \(=\) | \(\ds -1\) |
which is an inconsistency.
Hence there is no such solution.
$\blacksquare$
Sources
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.1$ Introduction: Example $\text {(ii)}$