Definition of linear equation

Definition: In the linear equation $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b$, the constants $a_1, a_2, \ldots, a_n$ are called the coefficients of the equation.

Each coefficient $a_i$ multiplies the corresponding variable $x_i$. The coefficients determine the "weight" or contribution of each variable to the equation.

• Coefficients can be any real (or complex) numbers: positive, negative, zero, integers, or fractions.
• A coefficient of $0$ means that variable does not appear in the equation.
• A coefficient of $1$ is typically not written explicitly (e.g., $x + y = 1$ means $1x + 1y = 1$).

Example: In $3x - \frac{1}{2}y + 0z = 7$, the coefficients are $3$, $-\frac{1}{2}$, and $0$ for $x$, $y$, and $z$ respectively.