Welcome to a lesson
on parallel and perpendicular lines.
Parallel lines are two lines that never intersect.
Notice here we have the graph
of two blue parallel lines.
If we look at the slopes of these two lines,
notice how the slope of this line
is equal to 1/3.
The slope of this line here is also equal to 1/3.
And this is true for all parallel lines.
The slopes of parallel lines are always equal.
And we can use this property
to help us determine equations of parallel lines.
In example one, we're asked to give the equation
of the line passing through the point (8,3)
that is parallel to the line y equals
negative two x plus three.
So the equation of the line we're trying to determine
is parallel to y equals negative two x plus three,
because the equation of this line
is in slope-intercept form, notice how
the slope of this line is negative two.
So because the slope of this line is equal to
negative two, and our line is parallel to this line,
we're trying to determine the equation of the line
with a slope that's also negative two,
and that passes through the point (8,3).
So because we know the slope of our line,
we know the equation must be y equals
negative two x plus b, and because this point
is on the line, this ordered pair
must satisfy the equation, so we can determine
the vertical intercept b, by performing substitution.
We'll substitute eight for x,
three for y, and then find b.
So we would have three equals negative 2
times x, which is eight, plus b.
Simplifying, we have three equals negative 16
plus b, adding 16 to both sides,
and now simplifying,
we have 19 equals b.
So now we know the slope is equal to negative two,
which we already have in our equation here.
And now we also know that the vertical intercept b
is equal to 19.
Which means the equation of our line
is y equals negative two x plus 19.
This line is parallel to y equals negative two
x plus three, and passes through the point (8,3).
Now let's talk about perpendicular lines.
Perpendicular lines pictured here
are two lines that intersect and form a right angle,
where a right angle measures 90 degrees.
If we take a look at the slope of this first
line here, notice how it would be negative three over one.
And the slope of this line is positive 1/3.
The slopes of perpendicular lines
have a special relationship.
They are always opposite reciprocals,
also referred to as negative reciprocals.
To find the opposite reciprocal
of a number, we put the number in fraction form,
flip it over, and change the sign.
So again, the slopes of perpendicular lines
are opposite reciprocals.
And now let's complete this table for practice.
We're told that line one and line two
are perpendicular, and we're given the slope
of line one, and asked to find the slope of line two.
So all these slopes here will be
the opposite reciprocals of the given slopes.
So if line one has a slope of 2/3,
the opposite reciprocal would be negative 3/2.
Again, we flip it over and change the sign.
If the slope of line one is five,
or five over one, the opposite reciprocal
would be negative 1/5.
If the slope of line one is negative eight,
or negative eight over one, we flip it over
and change the sign.
The opposite reciprocal is positive 1/8.
And if the slope is negative 4/5,
the opposite reciprocal would be positive 5/4.
Let's look at another example.
We want to give the equation of the line
passing through the point (8,3)
that is perpendicular
to the line y equals negative two x plus three.
For this problem, our line is going
to be perpendicular to this line.
Well, notice how the slope of this line
would be equal to negative two,
or negative two over one.
And because our line is perpendicular to this,
the slope would be the opposite reciprocal,
so if you flip it over and change the sign,
our slope is going to be positive one over two.
So we're trying to find the equation
of a line with a slope of 1/2,
that passes through the point (8,3).
Well, because we know the slope, now,
we know the equation would have to be
in the form of y equals 1/2 x plus b.
And now using the point that we know
is on the line, and therefore,
this ordered pair satisfies the equation,
we'll perform substitution to find
the vertical intercept.
So we'll substitute eight for x,
and three for y, just as we did before,
but now the slope is different.
So performing the substitution,
we would have three equals 1/2
times x, which is eight, plus b.
Simplifying, we have three equals
1/2 times eight is equal to four, plus b.
Subtracting four on both sides.
Simplifying, three minus four is negative one.
Negative one equals b.
So now that we know,
now that we know that the perpendicular line
has a slope of positive 1/2, and a vertical
intercept of negative one, we can now
write the equation for this line.
The equation is going to be y equals
1/2 x and then because b is equal to negative one,
we'd have minus one.
I hope you found this helpful.