Parallel and Perpendicular Lines
One application of slope involves deciding whether two lines
are parallel. Since two parallel lines are equally
“steep”, they should have the same slope. Also, two
lines with the same “steepness” are parallel.
Parallel Lines
Two lines are parallel if and only if they have the same
slope, or if they are both vertical.
Example
Parallel Line
Find the equation of the line that passes through the point
(3, 5) and is parallel to the line 2x + 5y = 4.
Solution
The slope of 2x + 5y = 4 can be found by writing the equation
in slopeintercept form.
2x + 5y = 4
This result shows that the slope is -2/5. Since the lines are
parallel, -2/5 is also the slope of the line whose equation we
want. This line passes through (3, 5). Substituting m = -2/5, x1
= 3 and y1 = 5 into the point-slope form gives
y - y1 = m(x - x1)
As already mentioned, two nonvertical lines are parallel if
and only if they have the same slope. Two lines having slopes
with a product of -1 are perpendicular.
Perpendicular Lines
Two lines are perpendicular if and only if the product of
their slopes is - 1, or if one is vertical and the other
horizontal.
Example
Perpendicular Line
Find the slope of the line L perpendicular to the line having
the equation 5x - y = 4.
Solution
To find the slope, write 5x - y = 4 in slope-intercept form:
y = 5x - 4
The slope is 5. Since the lines are perpendicular, if line L
has slope m , then
5m = -1
Many real-world situations can be approximately described by a
straight-line graph. One way to find the equation of such a
straight line is to use two typical data points from the graph
and the point-slope form of the equation of a line.
