Simultaneous Linear Equations/Examples/Arbitrary System 2

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 $\text {(ii)}$