Coordinate geometry or Analytic geometry
     Line in a coordinate plane
      Slope-intercept form of a line,  y = mx + c
         Slope or gradient, y-intercept and x-intercept
      The intercept form of the equation of the line
         Lines parallel to the coordinate axes, horizontal and vertical lines
      The point slope form of the equation of a line
         The two point form of the equation of a line
Line in a coordinate plane
Slope-intercept form of the equation of a line
  m   is the slope,
  x0  is the x-intercept,
   c   is the y-intercept.
The points at which the line  y = mx + c  intersects the coordinate axes are denoted as, x0 and c, in the picture above.
The intercept form of the equation of a line
We use the slope intercept form 
to get the intercept form of the equation of the line (see the figure above).
Lines parallel to the axes, horizontal and vertical lines
If  m = 0, to every number x associated is the same constant value y = c. A line parallel to the x-axis is called a horizontal line (or constant).
If the x value never changes a line is parallel to the y-axis. A line parallel to the y-axis is called a vertical line.
The point slope form of the equation of a line
The equation of a line that passes through the given point (x1, y1) and has the given slope m is represented by the definition of the slope and is called point-slope form or the gradient form of the line.
Since the slope of a line is the ratio of its vertical change to its horizontal change then
or     y - y1 = m(x - x1)
The equation can also be considered as
y = mx  translated to the point P1(x1, y1).
Example:  Find the equation of the line that is parallel with the line y = - x - 2 and passes through the point P1( 2, 1) .
The two point form of the equation of a line
Two points P1(x1, y1) and P2(x2, y2) determine a unique line on the Cartesian plane, therefore their coordinates satisfy the equation y = mx + c
The equation of the line which passes through the point  P1(x1, y1)  is   y - y1 = m(x - x1).  As the point P2(x2, y2) lies on the same line, its coordinates must satisfy the same equation, so  y2 - y1 = m(x2 - x1).
Thus, the slope
then
is the equation of the line passing through the two points. 
Example:  Find the equation of the line which passes through points P(-2, 3) and Q(6, -1).
Coordinate geometry contents
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