

Equations
of straight line

Slope of a line 
Slopeintercept form
of a line 
The pointslope 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 xaxis. 
The slope
is the difference quotient 





Slopeintercept form
of a line 


The
pointslope form of a line

The
equation of a line that passes through the given point (x_{1},
y_{1})
and has the given slope m
is represented by the
definition of the slope and is called pointslope 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

y_{1 }= m(x

x_{1}) 
The
equation can also be considered as the translation of
the source linear function y
= mx to the
point P_{1}(x_{1},
y_{1}). 



Example:
Find the equation of the line that is parallel with the line y
=  x
 2 and passes through the point 
P_{1}( 2,
1) . 


The two point form of
the equation of a line 
Two
points P_{1}(x_{1},
y_{1})
and P_{2}(x_{2},
y_{2})
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 P_{1}(x_{1},
y_{1})
is y 
y_{1}
= m(x
 x_{1}).
As the point 
P_{2}(x_{2},
y_{2})
lies on the same line, its coordinates must satisfy the same
equation, so y_{2}
 y_{1}
= m(x_{2}
 x_{1}). 
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 m_{1}
and m_{2
}are parallel if 

m_{1}
= m_{2} 

that
is, if they have the same slope. 


To
acquire the criteria when two lines, y
= m_{1}x
and y
= m_{2}x 
are
perpendicular or orthogonal we can
use the principle of similar triangles, OA'A
and OB'B
in the picture. 
Therefore, 
m_{1
}: 1 =
1_{
}: m_{2} 
=> 






This
relation will stay unchanged if we translate the perpendicular
lines, that is, when lines 
y = m_{1}x
+ c_{1
}and y
= m_{2}x
+ c_{2
}are written in the slopeintercept 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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