What does a vector solution to a system of linear equations represent

• A vector $\vec{s}$ is a solution to a system of $m$ equations in $n$ variables if all $m$ equations are satisfied when $x_i=s_i$ for $i=1,...,n$.

Eg:

$$\begin{array}{r}2x_1-2x_2+2x_3=3\\ x_1-5x_3=0\\ \\ \vec{s}=\left[\begin{array}{c}0\\ -\frac{3}{2}\\ 0\end{array}\right]\end{array}$$

Subbing the respective $s_i$ for the respective $x_i$ we can see that both the equations are satisfied.

What does it represent

• The system of linear equations here represents two planes ($x_3$).
• $\vec{s}$ represents a point where the two planes intersect.