Welcome to the presentation on finding the equation of a line.
Let's get started.
Say I had two points.
Let's say I have the point 1 comma 2, and I have the point 3
comma 4, and I want to figure out the equation of the
line through these points.
So let's at least figure out what that line looks like.
So 1 comma 2 is here, and 2, 3; 3, 4.
3 comma 4 is here, and if I want to draw a line
through them, it'll look something like that.
So what we want to do is figure out the equation of this line.
Well, we know the form of an equation of a line is y equals
mx plus b, where m is the slope, and that tells you how
steep the line is, and b is the y-intercept.
And I don't know why people chose m and b.
We'll have to do some research on that.
b is the y-intercept and the y-intercept is just where does
it intersect the y-axis.
And this problem, you could actually look at it and
figure it out, but let's do it mathematically.
So the equation for the slope m: it's rise over run.
Another way to view that is for any amount that you
run along the x-axis, how much do you rise?
Well, let's do that numerically.
Rise is the same thing as change over y, and run is the
same thing as change over x.
Delta, this triangle, means change, change in y.
Well, change in y, let's take the starting
point to be 3 comma 4.
Let's say we're going from 3 comma 4 to 2 comma 1.
The change in y is 4 minus 2.
We just took this 4 minus this 2 over 3 minus 1.
My phone was ringing.
And that's just this 3 minus this 1.
So if we just solve for it, we get 4 minus 2 is 2, and 3
minus 1 is also 2, so we get the slope is equal to 1.
And that makes sense because when we move over 1 in x,
we go up exactly 1 in y.
When we move to the left 1 in x, we move down exactly 1 in y.
So now we know the equation is y equals 1x plus b because
we solved the m equals 1.
And this is, of course, the same thing as
y equals x plus b.
Now, all we have left to do is solve for b.
Well, how do we do that because we have three variables here.
Well, we could actually substitute one of these pairs
of points in for y and x, and that makes sense, because
these points have to satisfy this equation.
So let's take this first pair. y is equal to 2.
2 is equal to x, which is 1 plus b.
It's a pretty easy equation to solve.
We get b equals 1, so that tells us that the equation of
this line is y equals x plus 1.
That's a pretty straightforward equation, and it makes sense.
The y-intercept is 1, which is exactly here, 0 comma
1, and the slope is 1, and that's pretty obvious.
For every amount that we move to the right, we move the same
amount up, so the slope is 1.
Let's do another problem.
Let's say I wanted to find the equation of the line between
the points negative 3 comma 5 and 2 comma negative 6.
Well, we do the same thing.
m is equal to change in y over change in x.
So let's take this as the starting point.
So say negative 6 minus 5.
So we just took negative 6 minus 5 over 2
minus negative 3.
You've got to be real careful to get the signs right.
So it's 2 minus negative 3.
Negative 6 minus 5 is minus 11, and 2 minus negative 3, well,
that's the same thing as two plus plus 3, so that's 5.
So we have the slope is equal to negative 11/5.
And notice that if on the numerator we use negative 6 as
the starting point, that in the denominator, we have to use
2 as the starting point.
We could have done it the other way around.
We could have said 5 minus negative 6 over negative 3
minus 2, in which case we would have gotten-- this would have
been 11 over negative 5.
So as long as you-- if you use the negative 6 first, then you
have to use the 2 first, or if you use the 5 first, then
you have to use the negative 3 first.
I hope I'm not completely confusing you guys.
Well, anyway, we know the slope is negative 11/5, so the
equation of this line so far is y equals minus 11/5x plus b.
Now we can take one of these pairs on the top and substitute
back and solve for b.
Let's take the first pair.
So 5 is y.
So we say 5 equals negative 3, so it's negative 11/5
times negative 3, right?
I just put the x in for x plus b.
So just simplifying that, I get 5 is equal to 33/5 plus b, or b
is equal to 5 minus 33/5, and this equals 25 minus 33/5.
25 minus 33 is minus 8/5.
So the equation of this line, and this one wasn't as clean as
the other one, obviously, is-- let me do it in another color
for emphasis-- y equals minus 11/5x minus 8/5.
Hopefully, those two examples will give you enough of an idea
to do the figuring out the equation of a line problems.
And if you have problems with this, you might just want to
try just the slope of the line problems or the y-intercept
problems separately.
I hope you have fun.
Bye.