

Graphing
Linear Equation, Linear Function (First Degree Polynomial)

The
pointslope 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
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) . 


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). 


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 slopeintercept 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 xaxis. 


And setting
B = 0 gives 

the
line parallel to the yaxis. 



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). 










Intermediate
algebra contents 



Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 