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== Equation of 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.
 
===Slope–intercept form or Gradient-intercept form===
A non-vertical line can be defined by its slope {{mvar|m}}, and its {{mvar|y}}-intercept ''y''<sub>0</sub> (the {{mvar|y}} coordinate of its intersection with the {{mvar|y}}-axis). In this case its ''linear equation'' can be written 
:<math>y=mx+y_0.</math>

If, moreover, the line is not horizontal, it can be defined by its slope and its {{mvar|x}}-intercept ''x''<sub>0</sub>. In this case, its equation can be written
:<math>y=m(x-x_0),</math>
or, equivalently,
:<math>y=mx-mx_0.</math>

These forms rely on the habit of considering a non vertical line as the graph of a function. For a line given by an equation 
:<math>ax+by+c = 0,</math>
these forms can be easily deduced from the relations 
:<math>\begin{align}
m&=-\frac ab,\\
x_0&=-\frac ca,\\
y_0&=-\frac cb.
\end{align}</math>

===Point–slope form or Point-gradient form===

A non-vertical line can be defined by its slope {{mvar|m}}, and the coordinates <math>x_1, y_1</math> of any point of the line. In this case, a linear equation of the line is
:<math>y=y_1 + m(x-x_1),</math>
or 
:<math>y=mx +y_1-mx_1.</math>

This equation can also be written
:<math>y-y_1=m(x-x_1)</math>
for emphasizing that the slope of a line can be computed from the coordinates of any two points.

===Intercept form===
A line that is not parallel to an axis and does not pass through the origin cuts the axes in two different points. The intercept values ''x''<sub>0</sub> and ''y''<sub>0</sub> of these two points are nonzero, and an equation of the line is
:<math>\frac{x}{x_0} + \frac{y}{y_0} = 1.</math>
(It is easy to verify that the line defined by this equation has ''x''<sub>0</sub> and ''y''<sub>0</sub> as intercept values).

===Two-point form===
Given two different points (''x''<sub>1</sub>, ''y''<sub>1</sub>) and (''x''<sub>2</sub>, ''y''<sub>2</sub>), there is exactly one line that passes through them. There are several ways to write a linear equation of this line.

If ''x''<sub>1</sub> ≠ ''x''<sub>2</sub>, the slope of the line is <math>\frac{y_2 - y_1}{x_2 - x_1}.</math> Thus, a point-slope form is
:<math>y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1).</math>

By clearing denominators, one gets the equation 
:<math> (x_2 - x_1)(y - y_1) - (y_2 - y_1)(x - x_1)=0,</math>
which is valid also when ''x''<sub>1</sub> = ''x''<sub>2</sub> (for verifying this, it suffices to verify that the two given points satisfy the equation).

This form is not symmetric in the two given points, but a symmetric form can be obtained by regrouping the constant terms:
:<math>(y_1-y_2)x + (x_2-x_1)y + (x_1y_2 - x_2y_1) =0</math>
(exchanging the two points changes the sign of the left-hand side of the equation).

===Determinant form===
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.

The equation <math> (x_2 - x_1)(y - y_1) - (y_2 - y_1)(x - x_1)=0</math> is the result of expanding the determinant in the equation
:<math>\begin{vmatrix}x-x_1&y-y_1\\x_2-x_1&y_2-y_1\end{vmatrix}=0.</math>

The equation <math> (y_1-y_2)x + (x_2-x_1)y + (x_1y_2 - x_2y_1)=0</math> can be obtained be expanding with respect to its first row the determinant in the equation
:<math>\begin{vmatrix}
x&y&1\\
x_1&y_1&1\\
x_2&y_2&1
\end{vmatrix}=0.</math>

Beside being very simple and mnemonic, this form has the advantage of being a special case of the more general equation of a hyperplane passing through {{mvar|n}} points in a space of dimension {{math|''n'' – 1}}. These equations rely on the condition of linear dependence of points in a projective space.

== Licensing == 
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Linear_equation Linear equation, Wikipedia] under a CC BY-SA license

== Resources ==
* [https://www.youtube.com/watch?v=gvwKv6F69F0 Equation of a Line], Khan Academy
* [https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb11595b61c86:writing-slope-intercept-equations/v/equation-of-a-line-1 Slope-intercept equation from slope & point], Khan Academy