section_path: Definition/Linear equation
section_name: Linear equation
paragraph_text: Linear equation
  
  Lines in a Cartesian plane or, more generally, in affine coordinates, are characterized by linear equations. More precisely, every line L {\displaystyle L} (including vertical lines) is the set of all points whose coordinates (x, y) satisfy a linear equation; that is, L = { (x, y) ∣ a x + b y = c }, {\displaystyle L=\{(x,y)\mid ax+by=c\},} where a, b and c are fixed real numbers (called coefficients) 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 c = 1 or c = 0, by dividing everything by 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 a x + b y − c = 0, {\displaystyle ax+by-c=0,} 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, x-intercept, known points on the line and y-intercept.
  
  The equation of the line passing through two different points P 0 (x 0, y 0) {\displaystyle P_{0}(x_{0},y_{0})} and P 1 (x 1, y 1) {\displaystyle P_{1}(x_{1},y_{1})} may be written as (y − y 0) (x 1 − x 0) = (y 1 − y 0) (x − x 0). {\displaystyle (y-y_{0})(x_{1}-x_{0})=(y_{1}-y_{0})(x-x_{0}).} If x ≠ x, this equation may be rewritten as y = (x − x 0) y 1 − y 0 x 1 − x 0 + y 0 {\displaystyle y=(x-x_{0})\,{\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}+y_{0}} or y = x y 1 − y 0 x 1 − x 0 + x 1 y 0 − x 0 y 1 x 1 − x 0. {\displaystyle y=x\,{\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}+{\frac {x_{1}y_{0}-x_{0}y_{1}}{x_{1}-x_{0}}}\,.} In two dimensions, the equation for non-vertical lines is often given in the slope–intercept form:
  
  y = m x + b {\displaystyle y=mx+b} where:
  
  The slope of the line through points A (x a, y a) {\displaystyle A(x_{a},y_{a})} and B (x b, y b) {\displaystyle B(x_{b},y_{b})}, when x a ≠ x b {\displaystyle x_{a}\neq x_{b}}, is given by m = (y b − y a) / (x b − x a) {\displaystyle m=(y_{b}-y_{a})/(x_{b}-x_{a})} and the equation of this line can be written y = m (x − x a) + y a {\displaystyle y=m(x-x_{a})+y_{a}}.
  
  As a note, lines in three dimensions may also be described as the simultaneous solutions of two linear equations a 1 x + b 1 y + c 1 z − d 1 = 0 {\displaystyle a_{1}x+b_{1}y+c_{1}z-d_{1}=0} a 2 x + b 2 y + c 2 z − d 2 = 0 {\displaystyle a_{2}x+b_{2}y+c_{2}z-d_{2}=0} such that (a 1, b 1, c 1) {\displaystyle (a_{1},b_{1},c_{1})} and (a 2, b 2, c 2) {\displaystyle (a_{2},b_{2},c_{2})} are not proportional (the relations a 1 = t a 2, b 1 = t b 2, c 1 = t c 2 {\displaystyle a_{1}=ta_{2},b_{1}=tb_{2},c_{1}=tc_{2}} imply t = 0 {\displaystyle t=0}). This follows since in three dimensions a single linear equation typically describes a plane and a line is what is common to two distinct intersecting planes.