In this exercise we'll be determining
the equations of parallel lines and perpendicular lines.
So for a quick review,
parallel lines, pictured here,
are two lines that never intersect.
Parallel lines have the same slope,
but their vertical intercepts, or y-intercepts,
must be different,
because if they were the same,
we would have the same line twice
where the two lines would coincide.
Perpendicular lines, pictured here,
intersect each other and form a right angle,
or form an angle that measures 90 degrees.
Perpendicular lines have slopes
that are negative reciprocals of one another.
So we have five problems here.
Let's take a look at three of the five.
We first want to give the equation of the line
passing through the point
negative eight comma -29,
that is parallel to y equals three x minus one.
And since we're trying to find an equation
of a parallel line and the given line
is in slope intercept form,
notice how the given line has a slope of three,
and because we're looking for a parallel line,
the parallel line will also have a slope of positive three.
So the line we're looking for has a slope of three,
and it also passes through
the point negative eight comma -29.
So now let's find our equation using slope intercept form.
We'll notice how we know the slope equals three,
which means our linear equation must be
in the form y equals three x plus b.
But because this point is on the line,
the ordered pair must satisfy this equation,
which means we can substitute negative eight for x,
29 for y,
which will allow us to find b, the vertical intercept.
So performing this substitution,
we would have -29 equals three times negative eight plus b.
And now we can solve for b.
So we have -29 equals -24 plus b,
so to solve for b you would add 24 to both sides,
b equals negative five.
Now that we know the vertical intercept
we now know the equation of the parallel line.
The equation would be y equals three x minus five.
Let's also verify this graphically.
The given equation is graphed in red,
and the equation we found is graphed in blue.
Notice how the two lines are parallel.
Next we want to give the equation of the line
passing through the point eight comma negative two
that is perpendicular to y equals 4/5x plus four.
Well now we're looking for an equation of a line
that's perpendicular to the given line
and because the given line has a slope of 4/5,
our slope is going to be the negative reciprocal of 4/5,
which is -5/4.
So we're looking for the equation of a line
that has a slope of -5/4.
Notice how to find this negative reciprocal.
We flip the fraction over and change the sign.
And the line also passes through the point
eight comma negative two.
So again using slope intercept form,
since we know the slope
we know the equation must be in the form
y equals negative 5/4x plus b.
This point is on this line
and therefore the ordered pair must satisfy the equation.
So now we'll substitute eight for x and negative two for y
which will allow us to find b, the vertical intercept.
So we would have negative two equals negative 5/4
times eight which I'll write as 8/1,
since our slope is a fraction, plus b.
So we have negative two equals,
let's simplify here.
There's one four in four and two fours in eight,
so we have negative five times two,
that's -10 plus b.
Adding 10 to both sides we have b equals positive eight.
Which means the equation of our perpendicular line
passing through the given point
would be y equals -5/4x plus eight.
Let's also check this graphically.
The given line is graphed in red,
the given point is plotted here in red,
and we found the equation to this blue line,
which you notice is perpendicular to the red line.
So this does verify our work is correct.
And we have one more problem.
We want to give the equation of the line
passing through the point three comma negative seven
that is parallel to y equals negative three.
So here notice how our equation y equals negative three
is going to be a horizontal line
and therefore we're looking for the horizontal line
that passes through the point three comma negative seven,
which would be y equals negative seven.
We'll verify this graphically in just a moment.
The other approach would be to recognize
that y equals negative three is a horizontal line
that has a slope of zero.
And because we're looking for a line that's parallel
to this the parallel line would also have a slope of zero.
So we're looking for a line that has a slope of zero
and passes through the point three comma negative seven.
Following the same procedure as before,
in slope intercept form our equation must be
in the form of y equals zero times x plus b.
Of course zero times x is just zero.
But we know this point is on our line
and therefore this ordered pair must satisfy our equation,
so we could substitute three for x
and negative seven for y.
If we do this we would have negative seven
equals zero times three plus b.
Notice that we have negative seven equals b,
so our simplified equation as we said before
is just y equals, this is zero,
b is negative seven,
so we have y equals negative seven.
Again it's much faster if we recognize that
any horizontal line passing through this point here
must be y equals negative seven.
Again let's verify this graphically.
The given line is graphed in red.
We were looking for the line parallel to the red line
passing through this red point
where this point has coordinates three negative seven
and therefore the parallel line here
is y equal negative seven, as we found.
I hope you found this helpful.