Equations of straight line
      Slope of a line
      Slope-intercept form of a line
      The point-slope form of a line
      The two point form of the equation of a line
      Parallel and perpendicular lines
Equations of straight line
Slope of a line
The slope of a line is the ratio of its vertical change to its horizontal change, or it is the tangent of the angle between the direction of the line and the x-axis.
The slope is the difference quotient  
Slope-intercept form of a line
 
The point-slope form 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 the translation of the source linear function y = mx  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).
   
Parallel and perpendicular lines
Two lines having slopes m1 and m2 are parallel if 
  m1 = m2   that is, if they have the same slope.  
To acquire the criteria when two lines,  y = m1x and y = m2x
are perpendicular or orthogonal we can use the principle of similar triangles, OA'A and OB'B in the picture. 
Therefore, m1 : 1 = -1 : m2 =>
 
This relation will stay unchanged if we translate the perpendicular lines, that is, when lines 
              y = m1x + c1 and  y = m2x + care written in the slope-intercept form.
Two lines are perpendicular if the slope of one line is the negative reciprocal of the other.
Example:  Find the equation of the line that is perpendicular to the line   and passes through 
                  the point A(-2, 5).
College algebra contents C
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