Welcome to a lesson on the slopes of parallel and perpendicular lines. Let's
begin by talking about parallel lines. Parallel lines or two or more lines in a
plane that never intersect. Examples of parallel lines are all around, such as
the opposite sides of a rectangular picture frame and the shelves of a
bookcase. So here's a graph of two parallel lines. Two non vertical lines in a
plane are parallel if they have both the same slope and different y intercepts.
The reason they must have different y intercepts is because it two lines have
the same slope and the same y intercepts, they would be the same line.
Let's look at a couple of examples. Here we're asked to find the slope of a line that is
parallel to the line y equals negative two-thirds x pls 2. Notice how the given
equation is in slope intercept form, which is the form y equals m x plub b, where m
is a slope and b is the y intercept.
Notice how the given line has a slope of negative two-thirds which means the
slope of a parallel line must also be negative two-thirds.
Let's look one example of a parallel line to the given line.
The blue line is the graph of y equals negative two-thirds x plus two. The red line is
just one example of a line parallel to the given line. The red line has an
equation y equal negative two-thirds x minus two. Notice how the slopes are equal
and the y intercepts are different. In this example we are asked to determine whether the
lines y equals 2 x minus 5 and y equals 2 x plus 1 are parallel.
Again both equations are in slope intercept form. Notice how both lines have a slope of 2.
The first line has a y intercept of negative 5. The second line has a y intercept of 1.
The slope are the same and the y intercepts are different. The lines are parallel.
Let's verify this graphically. Here's the graph of the two lines. And we can see
two lines are parallel. Now let's talk about about the slopes of perpendicular lines.
Perpendicular lines or two or more lines that intersect at a 90 degree angle or a
right angle. And here's an example of two lines that are perpendicular.
Again they intersect and form a 90 degree angle or a right angle. Two non vertical lines are
perpendicular if the slope of one line is the negative reciprocal of the slope of the
other. So for example if one line has a slope of a over b,
If the second line is perpendicular, it would have to have a slope of negative b over a.
So to find a negative reciprocal we flip the fraction over and change the sign.
Let's look at an example.
We're ask to find the slope of a line that is perpendicular to the line y equals
negative 3 x plus 2. Notice the slope of the given line ins negative 3.
So if a line is perpendicular to this line the slope must be the negative reciprocal
of negative 3. To find the negative reciprocal let's first
write this is a fraction which should be negative three over 1. So the negative
reciprocal would have to be positive one over 3, which means the slope of a
perpendicular line is equal to one-third.
Let's look at one example of a line that is perpendicular to the given line. The blue line is the
graph of y equals negative three x plus two. Notice of red line is perpendicular
to the blue line. The two lines intersect and form a 90 degree angle or right angle.
The equation the red line is y equals one third x minus 3. Notice how the slopes are
negative reciprocals of one another. For our last example we are asked to determine whether the lines
y equals 4 x minus 6 and y equals negative one fourth plus 1 are parallel,
perpendicular or neither. Looking at the equation of the first line notice how the
slope m is equal to 4. Let's call it m sub 1 for the first line. So m sub 1 is equal to 4.
The vertical intercept is negative six. Let's say b sub 1 equals negative 6.
Now for the second line the slope is negative one fourth. So it's called m sub 2
equals negative one fourth. And the vertical intercept is equal to 1. So we'll say b sub 2 is
equal to 1. The two slopes aren't equal so we know the lines are not parallel. If they're
perpendicular the slopes must be negative reciprocals of one another. If we can
write m sub 1 as m sub 1 equals 4 over 1, notice 4 over 1 and negative one fourth
are negative reciprocals and therefore the two lines are perpendicular.
Let's take a look at the graph
of the two lines. y equals 4 x minus 6 is graphed in blue. y equals negative one fourth x plus 1
is graphed in red. Notice how the two lines to do intersect and form a 90 degree angle or
right angle, verifying the two lines are perpendicular. I hope you found
this helpful.