System of Simultaneous Equations may have Unique Solution

Theorem

Let $S$ be a system of simultaneous equations.

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

Proof

Consider this system of simultaneous linear equations:

\(\text {(1)}: \quad\)

\(\ds x_1 - 2 x_2 + x_3\)

\(=\)

\(\ds 1\)

\(\text {(2)}: \quad\)

\(\ds 2 x_1 - x_2 + x_3\)

\(=\)

\(\ds 2\)

\(\text {(3)}: \quad\)

\(\ds 4 x_1 + x_2 - x_3\)

\(=\)

\(\ds 1\)

From its evaluation it has the following unique solution:

\(\ds x_1\)

\(=\)

\(\ds -\dfrac 1 2\)

\(\ds x_2\)

\(=\)

\(\ds \dfrac 1 2\)

\(\ds x_3\)

\(=\)

\(\ds \dfrac 3 2\)

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