Definition of coefficients of a linear equation

Definition: A solution to a linear equation $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$ is an ordered $n$-tuple of numbers $(s_1, s_2, \ldots, s_n)$ such that when each $x_i$ is replaced by $s_i$, the equation holds true:

$$a_1 s_1 + a_2 s_2 + \cdots + a_n s_n = b$$

In other words, substituting the solution values into the equation produces a valid equality.

Example: For the equation $2x + 3y = 6$:

• $(3, 0)$ is a solution because $2(3) + 3(0) = 6$ ✓
• $(0, 2)$ is a solution because $2(0) + 3(2) = 6$ ✓
• $(1, 1)$ is not a solution because $2(1) + 3(1) = 5 \neq 6$ ✗

A single linear equation in two or more variables has infinitely many solutions.