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:

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

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:

$$a_1x_1+a_2x_2+a_3x_3+\ldots+a_nx_n=b$$

where:

• $a_1,a_2,\ldots,a_n$ are called the coefficients.
• $x_1,x_2,\ldots,x_n$ 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 $3x+3y=0$, for example, is equivalent to $x+y=0$.