Let's talk about lines.
Two of the most important skills regarding
lines are graphing a line and writing the
equation of a line.
This example will have us do both.
We'd like to sketch the graph of a line with
slope negative 2/3 passing through the point
1, negative 2.
We'll start by plotting that point 1,
negative 2, and then we'll use the slope to
determine a second point on the line.
So our slope here is negative 2/3.
Remember that slope can be thought of as the
rise over the run, or the change in y over the change in x.
We're going to use this to plot that second point.
So, negative 2/3, and we're going to put the
negative sign with the 2, so we have a negative rise.
We're going to go from our starting point
down two units, and then the run is positive
3, so we're going to then go to the right three units.
This will give us the second point on our line.
That is enough to graph the line, but let's
go ahead and find a third point.
Suppose that I like to put that negative
sign in the denominator with the 3.
So, now my rise is 2, and the run is negative
3, so I'm going to go to the left three units.
What you'll notice is that all of those
points are in a straight line,
either of two of those would do.
Now, let's talk about writing the equation of this line.
We're going to start with point-slope form,
and in point-slope form, the m stands for
the slope as usual, and then x1, y1 are
going to be the coordinates of any point on the line.
We can use the one we're given 1, negative 2, as that point.
So, let's substitute, y1 is negative 2, so
we have y minus a negative 2, the slope is
negative 2/3, and x1 is 1.
I'm going to go ahead and distribute on the right-hand side.
Then let's think about how our answer should look.
Standard form is A x plus B y equals C.
So, we need to start out by getting the x
and y terms on the left and put everything
else on the right-hand side.
Let's add 2/3 x to both sides.
I want to subtract this 2.
So, we have to do 2/3 minus 2.
Okay, that's 2/3 minus 6/3 or
negative 4/3, and we're very close.
It's customary in standard form for the
coefficients to be integers when possible.
We can do that here by multiplying
everything, each term all the way across, by 3.
So, that gives me 2 x plus 3 y equals negative 4.
This is the equation of the line in standard form.
Now, let's look at another example.
In this example, we're also looking to
sketch the graph of a line and write its equation.
Here, we look at two special types of lines however.
We want to sketch the graph and write an
equation for the line that passes through
the point negative 2, 3 and has an undefined slope.
Okay, so let me start by plotting the point negative 2, 3.
A line with an undefined slope is a vertical
line, so we want to draw the graph of a
vertical line that passes through the point negative 2, 3.
Notice that it has an x-intercept, this line, at negative 2.
So, the equation of this vertical line is
just x equals negative 2.
In this second part, we want to sketch the
graph and write an equation for the line
passing through the same point, negative 2, 3.
Now, this one has a slope equal to 0.
So, again, I plot my point, negative 2, positive 3.
Now, a line with a slope equal to 0 is a horizontal line.
So, I'm going to draw my horizontal line through this point.
And then, notice that this one crosses the y-axis at 3.
That's its y-intercept. So, the equation of the horizontal line
through that point is y equals 3.
Let's look at a few more examples.
This example asks us to write the
slope-intercept form of the equation of a line given two points.
To write the equation of a line, we always
want to start out with point-slope form.
In point slope form, we need to know the
slope, m, and the coordinates of one point
on the line to call x1 and y1.
In our example, we're given two different points.
So the first thing we need to do is find the slope of the line.
Let's do that here.
The slope of a line is the change in y over the change in x.
Given two points on the line, x1, y1 and x2, y2,
we can write the slope formula here and solve to find the slope.
Alright, so let's just make a choice.
I'm going to call this one x1, y1 and this one x2, y2.
y2 is negative 5 minus y1 which is 1 over x2
which is 2 minus x1 so I get a plus 2.
Okay, so this is negative 6/4 which is equal to negative 3/2.
Let me point out to you that I could have
chosen this to be x2, y2 and this to be x1, y1,
and I should have ultimately come up with the same slope.
Now I'm ready to use my point-slope form from before.
Alright, so I've called this y1, I'll go ahead and use that: y1 is 1.
The slope we found to be negative 3/2, and
x1 is negative 2, so that makes that x plus 2.
Alright, so I've written an equation for the line.
I want it to be in slope-intercept form.
This means that I want to solve for y.
First of all, let's distribute this.
So that's negative 3/2 x.
Negative 3/2 times positive 2 is going to
be, give me a minus 3 here, and I want to solve for y.
So, add 1 to both sides. That would give me
a minus 2, and this is in the form y equals m x plus b.
This is the slope, this is the y-intercept,
and that's your slope-intercept form.
And, again, let me point out that I could
have chosen this point to be the x2 and y2
and this one to be x1 and y1 and using the
other point, I should have still gotten the same equation.
In this example, we are again asked to write
the equation of a line, but as you'll see,
we're given slightly different information.
We'd like to write the equation in standard
form of the line passing through the point
negative 3, 2 and parallel to 2 x minus 4 y equals 5.
We know that to write the equation of a
line, we should almost always start with point-slope form.
And to use point-slope form, we need to know
the slope of our line, and we need to know
the coordinates of one point, x1, y1.
So we are given that point, and we can use this information.
We need to figure out the slope of our line.
We are given something that will help.
Our line should be parallel to 2 x minus 4 y
equals 5, and there's a relationship between the slopes of parallel lines.
Let's see if we can find the slope of the line.
What's a good way to do that?
One is to write it is in slope intercept
form, that means we want to solve for y and
have y equals m x plus b, so we can just pick out that slope.
Let's subtract 2 x from both sides and
then divide by negative 4.
So we have negative 2 over negative 4 is
positive 1/2 and then 5 over negative 4,
actually going to make this minus 5/4.
Okay, so the slope of this line is 1/2.
What's the relationship between the slopes of parallel lines?
Right, they're equal.
So, if the slope of our line is 1/2, the
slope of any line parallel to it is also ½.
That's the last piece of information we
needed, we can substitute into this equation
and write our line in the appropriate form.
y1 is two, our slope is 1/2, and we have x minus a negative 3.
Let's distribute the 1/2.
Okay, go back and look, we wanted this in
standard form, that's A x plus B y equals C,
so we are going to subtract the 1/2 x from both sides.
And we need to add 2 to the 3/2.
Okay, add it to both sides, 2 plus 3/2,
that's 4/2 plus 3/2, or 7/2.
It's customary for these coefficients to be
integers, it's also customary to write the
coefficient of x as a positive number in standard form.
We can accomplish both of those things by
multiplying each term by negative 2.
And this is the equation of our line written in standard form.
In this example, we again want to write the
equation of some lines, given some characteristics of those lines.
Here we want the equation of a line passing
through the point 3, 2 that is first
parallel to y equals 5, so let's draw a picture of this.
Here's the point 3, 2, and let's go ahead
and graph the line y equals 5.
Now this says that y is always 5, so for
every point on the line the y-value, the y-coordinate, has to be 5.
This must be a horizontal line, and of
course it'll have to cross the y-axis at 5
because every point must have a y-coordinate of 5.
Okay, the line we're interested in is
supposed to be parallel to y equals 5, so I
know right away that the line I want needs
to also be horizontal, but it has to pass through this point.
Let's draw a horizontal line that passes through this point.
What does the y-coordinate of every point on this line have to be?
Two, that's right, so the equation of this line must be y equals 2.
Let's look at the second part of the question.
We again want the equation of a line passing
through the point 3, 2, but this time we
want our line to be perpendicular to y
equals 5, so again let me plot the point 3,
2, and graph the horizontal line y equals 5.
Alright, the line we're interested in needs
to be perpendicular to y equals 5, so yes,
it has to be a vertical line.
We want this vertical line to pass through the point 3, 2.
Now, what's special about this line?
Well for one thing, the x-coordinate of
every point on the line has to be 3.
And that's what's going to give us the equation of our line.
Since the x-value is always 3 no matter the
y-value, the equation of the line is x equals 3.
Be sure to practice all of these skills
regarding lines so that you are comfortable doing the problems.