Definition of system of linear equations

Definition: A solution to a system of linear equations is an ordered $n$-tuple $(s_1, s_2, \ldots, s_n)$ that satisfies every equation in the system simultaneously.

That is, when each variable $x_i$ is replaced by $s_i$, all equations in the system become true equalities.

Example: For the system:
$$\begin{cases}\n2x + y = 5 \\\nx - 3y = -1\n\end{cases}$$

$(2, 1)$ is a solution because:

• $2(2) + 1 = 5$ ✓
• $2 - 3(1) = -1$ ✓

$(1, 3)$ is not a solution because $2(1) + 3 = 5$ ✓ but $1 - 3(3) = -8 \neq -1$ ✗.

A system may have zero solutions (inconsistent), exactly one solution (unique), or infinitely many solutions.