A system of linear equations (on linear systems) is a collection of one or more equations involving the same variables. For example,
$$ \begin{cases}3x+2y-z=1\\2x-2y+4z=-2\\-x+{\frac {1}{2}}y-z=0\end{cases} $$
is a system of three equations in the three variables $x$, $y$ and $z$.
In mathematics, a linear equation is an equation in which the degree of every variable is 1 → no variable in the linear equation has an exponent greater than 1. Therefore, it follows the form:
$$ a_1x_1 + a_2x_2 +... + a_nx_n + b = 0 $$
where $x_1,...,x_n$ are the variables, $a_1,...,a_n$ are coefficients and $b$ is a constant. Note that $b, a_1,...,a_n$ are all scalars (real numbers).
Conversely, nonlinear equations have variables with degrees other than 1.
Examples of Linear Equations
$$ a + b = 10 \\ 2a + 3b = 15 \\ 3.4a - 48.99b + 2c = 122.5 $$
Examples of Nonlinear Equations
$$ a^2 + b^2 = 10 \\ \sin (a) + 3b = 15 \\ 2^a - 3^b = 0 $$
Linear equations with 2 variables produce straight lines when graphed on a two-dimensional plane. The solution set of the system is the intersection of these lines.
Types of Linear Systems