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 $$x+y=0$$.