Linear equation

The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true.

The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line.

There are various ways of defining a line. In the following subsections, a linear equation of the line is given in each case.

for emphasizing that the slope of a line can be computed from the coordinates of any two points.

This form is not symmetric in the two given points, but a symmetric form can be obtained by regrouping the constant terms:

(exchanging the two points changes the sign of the left-hand side of the equation).

The two-point form of the equation of a line can be expressed simply in terms of a determinant. There are two common ways for that.

A linear equation with more than two variables may always be assumed to have the form

A solution of such an equation is a n-tuples such that substituting each element of the tuple for the corresponding variable transforms the equation into a true equality.