Editing Line (geometry) (section)

=== Linear equation ===
{{Main|Linear equation}}
[[File:Linear Function Graph.svg|alt=y = –x + 5 (going down) and y = 0.5x + 2 (rising up slower)|thumb|Line graphs of linear equations on the Cartesian plane ]]
Lines in a Cartesian plane or, more generally, in [[affine coordinates]], are characterized by linear equations. More precisely, every line <math>L</math> (including vertical lines) is the set of all points whose [[Cartesian coordinates|coordinates]] (''x'', ''y'') satisfy a linear equation; that is,
<math display="block">L = \{(x,y)\mid ax+by=c\}, </math>
where ''a'', ''b'' and ''c'' are fixed [[real number]]s (called [[coefficient]]s) such that ''a'' and ''b'' are not both zero. Using this form, vertical lines correspond to equations with ''b'' = 0.

One can further suppose either {{math|1=''c'' = 1}} or {{math|1=''c'' = 0}}, by dividing everything by {{mvar|c}} if it is not zero.

There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. The above form is sometimes called the ''standard form''. If the constant term is put on the left, the equation becomes
<math display="block">ax + by - c = 0,</math>
and this is sometimes called the ''general form'' of the equation. However, this terminology is not universally accepted, and many authors do not distinguish these two forms.

These forms are generally named by the type of information (data) about the line that is needed to write down the form. Some of the important data of a line is its slope, [[root of a function|x-intercept]], known points on the line and y-intercept.

The equation of the line passing through two different points <math>P_0( x_0, y_0 )</math> and <math>P_1(x_1, y_1)</math> may be written as
<math display="block">(y - y_0)(x_1 - x_0) = (y_1 - y_0)(x - x_0).</math>
If {{math|''x''<sub>0</sub> ≠ ''x''<sub>1</sub>}}, this equation may be rewritten as
<math display="block">y=(x-x_0)\,\frac{y_1-y_0}{x_1-x_0}+y_0</math>
or
<math display="block">y=x\,\frac{y_1-y_0}{x_1-x_0}+\frac{x_1y_0-x_0y_1}{x_1-x_0}\,.</math>In [[Plane (mathematics)|two dimensions]], the equation for non-vertical lines is often given in the ''[[slope–intercept form]]'':

<math display="block"> y = mx + b </math>
where:
* ''m'' is the [[slope]] or [[slope|gradient]] of the line.
* ''b'' is the [[y-intercept]] of the line.
* ''x'' is the [[independent variable]] of the function {{math|1=''y'' = ''f''(''x'')}}.

The slope of the line through points <math>A(x_a, y_a)</math> and <math>B(x_b, y_b)</math>, when <math>x_a \neq x_b</math>, is given by <math>m = (y_b - y_a)/(x_b - x_a)</math> and the equation of this line can be written <math>y = m (x - x_a) + y_a</math>.

As a note, lines in three dimensions may also be described as the simultaneous solutions of two [[linear equation]]s
<math display="block"> a_1 x + b_1 y + c_1 z - d_1 = 0 </math>
<math display="block"> a_2 x + b_2 y + c_2 z - d_2 = 0 </math>
such that <math> (a_1,b_1,c_1)</math> and <math> (a_2,b_2,c_2)</math> are not proportional (the relations <math> a_1 = t a_2, b_1 = t b_2, c_1 = t c_2 </math> imply <math>t = 0</math>). This follows since in three dimensions a single linear equation typically describes a [[plane (geometry)|plane]] and a line is what is common to two distinct intersecting planes.