Linear equation with n unknowns

Given the equation $x + y = 0$ it is said that it is a linear equation with $2$ unknowns $(x, y)$ and linear because there are not quadratic or higher terms.

This equation does not have a unique solution, meaning that there are more than one combination of values of $x$ and $y$ that satisfy the equation.

Possible solutions are: $(1, - 1), (2, - 2), (100, - 100)$ , etc.

The equation:

$x + y + 3 t - z = 2$

is also a linear equation, although now we have $4$ unknowns.

Obviously it does not have a unique solution either.

More generally, a linear equation with $n$ unknowns is defined as follows:

$a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} + \dots + a_{n} x_{n} = b$

where:

$a_{1}, a_{2}, \dots, a_{n}$ are called the coefficients.
$x_{1}, x_{2}, \dots, x_{n}$ are the unknowns.
$b$ is the constant term.

It is said, also, that two equations are equivalent when they have the same solution.

The equation $3 x + 3 y = 0$ , for example, is equivalent to $x + y = 0$ .