Hello again. So another 15 minutes on slopes
and equations of lines. We just got done talking about how slope is rise over run. That is
y sub 2 minus y sub 1 over x sub 2 minus x sub 1. We have some forms of equations of
lines. We have got Slope Intercept Form which is y equals mx plus b. We have Point Slope
Form which requires you to take a point and slope, the point is (X1,Y1) and the slope
is m. This is where we plug in our numbers when we find the equation of the line. We
also have Standard Form which is Ax plus By plus C equals 0. This is not very friendly
for graphing unless you are using the x and the y intercepts. We are going to pick back
up with some more examples on how to come up with some equations of lines given just
a minimum of information. The Point Slope form of a line tells us the absolute minimum
we need to write the equation of a line, a point x and y, and a slope. That is why it
asks for that information. In my last example we had… We were given two points, we found
the slope. We had to find our own slope and then we used the Point Slope form to find
the equation of the line. We are going to go right to that this time. Let’s say that
we have a slope of 2 and the point this time is not just going to be like (5,7). The point
is given to us as an x intercept. So my x intercept is equal to 4. So when I have a
point, I need an x and a y. But yet I said the x intercept is 4. Well, where is the x
and y? If you are crossing the x axis at 4… 1, 2, 3, 4… What is the coordinate of that
point? Well the coordinate of that point is 1, 2, 3, (4,0). So when I say the x intercept
if four, it only seems like you are given one number. I am actually giving you two.
You do have the x and the y coordinate. So below here we have a point again of (4,0).
That is going to be my x and my y. We are going to find the equation of the line. The
equation of the line. The Point Slope form says y-y1 so y minus the y value of the coordinate
is 0, equals m that is the slope I was given of 2, times x minus x sub one which is 4.
A lot of textbooks, and I won’t be happy if my kids give me an equation in this form,
because other than memorizing the Point Slope formula they have not done anything. So we
like to have these… I like to have these set equal to y. Because they are also very
easy to graph when y is by itself. So I am going to get rid of these parenthesis. We
are going to do that by taking the 2 and multiplying through. Subtracting zero does not do anything,
so I am not going to show that any more. y equals two times one is 2x, two times negative
four is -8. And BAM! I got my Slope Intercept form. y=mx+b So in case you are not very good
about graphing lines, let’s do that. This is my slope of 2 over 1. Now I am writing
this like a fraction because you need a rise value and a run value. Rise is either up or
it is down. It is up if it is positive, and mine is. And the bottom number I always read
graphs left to right. That is exactly a necessity, but I like to keep taking my up and down movement
as my positive movement or my negative movement. The bottom number always to the right. The
last number over here is my y intercept, my b. So b is equal to negative eight. When you
are graphing a line in slope intercept form, you do the y intercept first… I almost said
the wrong thing. You do the y intercept first of negative eight and then you do what the
slope says. Let’s see if we can graph it and get it to pass through this point like the
information said in the first place. It is in blue, so I will keep it in blue. The y
intercept is negative eight. Remember the y axis is the one that goes straight up and
down. So 1, 2, 3, 4, 5, 6, 7, 8. Now my slope is 2 over 1. So up 2 over 1, up 2 over 1,
up 2 over 1, and there is no reason for you to do this many repetitions. But look, I just
counted my way back to the point I was given in the first place. Pretty cool. So here is
my line. Y=2x-8. I have time for one more. Let’s take a look at… Lets do a weird one.
Lets say that my slope was undefined and the point that I am going through is… I don’t
know… (-2,4) Wow! I just told you that to find the equation of a line you needed a point
and a slope. I have a point, but my slope is undefined? How am I supposed to plug into
this equation… How am I supposed to plug into the Point Slope form if I don’t have
a slope? If you watched the last video, I kind of made some arm movements here trying
to show you some stuff. If you have a line going up to the right, that is a positive
slope. If it is falling down to the right, that is a negative slope. Horizontal lines,
that is a slope of zero. If you have a vertical line straight up and down, those lines have
an undefined slope. Unless you have that memorized there is nothing you can do with this question.
You are not going to be able to answer it. This means that you have a vertical line.
Vertical lines have the equation of x equals a number. In this case the x of the point
that the line is going through is negative two, so the equation is x=-2. That is it.
There are no x’s and y’s going on because the line is very simple. It is straight up.
The only thing changing is the y. I don’t have to know them. The only thing that I have
to know is that the x is staying a constant x value of negative two. If that is a special
case and it is pure memorization, what if I had a slope of zero? Well, if I have a slope
of zero and you don’t know the definition you can still work your way through the algebra
and get the right answer. But, if you know that… Ok, this is a slope of zero. Horizontal
lines have a slope of zero. Well, horizontal lines don’t have the equation x equals a number.
Horizontal lines have the equation y equals a number. So, y equals a number is the form
that we have for horizontal lines and what is the y value of this point that I am going
through. So take that number symbol out, y=4 and there is your answer. So there is two
examples. One in the old video and one in this one about finding equations of lines.
You need a minimum of a point and a slope. But there are these special cases. If you
don’t have these memorized, you are going to get those wrong. Let’s see here. I have
a few minutes left. Let’s just one more time take a look at graphing a line. We are going
to graph y equals 2x. Well slope intercept is y equals mx plus b. I have got the y by
itself so it is in slope intercept form. But it is… Oh. Well there is mx. There is nothing
over here. A lot of us will know what that means. But if you are just starting out in
Algebra you might not realize that there are telling you that the y value, the y intercept
excuse me, is zero. We don’t have to write plus zero because what is 5 plus 0? It is
still 5. It does not do anything. Well we have a slope of… You know what, let me make
this a negative. Negative two-thirds so yo can see that movement. So this line has a
slope which is negative 2/3. Now here is where I like to talk about your slope. It is negative.
I like to say that the top is rise up or a drop down. And this is going to go down because
the slope is negative. I like the bottom number is always to the right. Read those graphs
left to right. So we have a slope of negative two-thirds. We have a y intercept of b equals
zero. Again, one more time, when you are in slope intercept form you want to graph those
with the y intercept first and the slope second. So this first and this second. Here is your
line. Yes I do kind of write hard so I do break my chalk often. Y intercept is zero.
Done! Slope. Down two over three, down two over three. Or just to highlight it again,
down two over three to make my new point. There is my line. Let’s just say for a second
that this were y is greater than or equal to negative 2/3x. Now I have an inequality.
That means shading. Where to the y’s get bigger…greater than? This is y is greater than. Where to
y’s get bigger? When you go down? No, y’s get bigger as you go up. That is a y value
of 1, 2, 3, 4, so that inequality when you get to that in your Algebra, or Algebra 2
book, or Geometry will indicate that you shade up. Thank you very much. BAM!!! I am going
to BAM right back. Because I just realized that have a few more minutes to show you some
more stuff. I graphed using the slope intercept form twice. What about this equation here,
standard form? Well standard form is good if you are going to graph equations of lines
using x and y intercepts. Let’s talk about that. So here is an equation, 4x+3y+12=0.
I want a graph using x and y intercepts. If I don’t know what is happening, I am going
to erase this in a second, what is happening when you cross…when the line crosses the
x and y axis? If you are on the y axis, isn’t the x equal to zero. If I move from the origin,
this is up two. This point is (0,2). When you have a y intercept the x is zero. This
point over here that is on the x axis is (3,0). Three…Zero…so if I am on the x axis they
y value is zero. So y intercept, x equals zero. X intercept, y equals zero. I am going
to quicken this up just a little bit. I am going to bring the 12 over with subtraction.
4x+3y=-12. We are going to find the x intercept. You do that by letting the y equal zero. These
will fall away very quickly and you will have your answers. 4x plus 3 times 0 equals -12.
That is zero. 4x=-12. x when you divide both sides by four is negative three. Let’s find
one more point because you need two points to graph a line. My y intercept is when you
let x equal zero. That is going to be 4 times 0 plus 3y equals 12. I am almost out of space.
I am going to remind you that 4 times 0 is 0, and we are left with 3y=-12. When you divide
both sides by 3, -12 divided by 3…. I am undoing that multiplication… is going to
be negative four. So now I have two points. I have an x intercept and a y intercept. And
if I have two points I can graph a line. So can you:) We have an x y axis. We have an
x intercept of negative three. We have a y intercept of negative four. And through those
points we have a line. Now I am out of time. Thank you very much!!!