Graphing Linear Equation, Linear Function (First Degree Polynomial)
      The point-slope form of a line
      Parallel and perpendicular lines
      General form of the equation of a line
      The two point form of the equation 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) .
   
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 + cand  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).
General form of the equation of a line
The linear equation Ax + By + C = 0 in two unknowns, x and y, of which at least one of the coefficients, A or  B, are different then zero, is called the general form for the equation of a line.
Dividing the equation by B, where B is not 0, gives
  where  
the slope-intercept form  y = mx + c  of a line.
By putting C = 0 into the general form obtained is    
 the equation of the line that passes through the origin.
Setting A = 0, gives the line parallel to the x-axis.  
And setting B = 0 gives the line parallel to the y-axis.  
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).
   
Intermediate algebra contents
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