Today I'm going to show you yet another
reason why I keep telling you as soon as
you see a linear equation to begin by
solving for Y. Remember, the rectangular
coordinate system began with the x-axis
had 0, the positive numbers, and the
negative numbers to the left.
That's the x-axis. The y-axis of course went
vertically with the positive numbers
going up and the negative numbers going
down.
OK? Now, let's look at this line,
the line represented representing the
equation Y= 2X-1
That's it. You just have to trust me.
That point right there is where it's
intersecting the y-axis, isn't it?
Well that point is called its
y-intercept. That is just what it is. It's
where it intersects the y-axis
Let's look at another. This equation: y equals
-3x plus 4. Let's solve for y.
If I graphed it, it would be right here. Now where
is it intersecting the y-axis? Its
y-intercept is 4 because that's where
it intersects the y-axis. You might want
to note that its X intersept, intercept
is one-and-a-half but for right now
we're only talking about the y-intercept.
Now let's go back to this line. We know
the y-intercept is minus 1. I want you to
note something. What is the value of x at
the y intercept? In other words where are
you on the X line? Well, it's right in the
middle, isn't it? It's 0 and if you put 0
into this equation for x. Of course, the part
of it will cancel out and you'll get the
constant y equals minus 1 well that was
the y-intercept, wasn't it? The point 0-1
We're going to have a pattern here. We go
back to the second equation and put in 0
for X, what are we going to be left with?
The constant so you could say if you'd
solve for y that the number, the constant
sitting by itself in that equation will
always be the y intercept, the other
number, the number in front of x will
always be the slope. You don't have to go
to any trouble if you've solved for y.
You don't have to go to any trouble of
subtracting and dividing and all that if
you'll solve for y that number is always
the slope. So in summary
if we solve an equation for y, it is in
what's called slope-intercept form. It
makes sense doesn't it?
Where the number in front of X is always
the slope and the other number is always
the y-intercept. For instance, in this
equation, now that I've solved for y the
slope is seven and the y-intercept is
three. Just that easy! In this equation,
the y-intercept is seven and the slope
is minus 2 and note I said the number in
front of X is the slope not the first
number, the number in front of x.
So if I switch it negative 2 is still
the y-intercept and six is still the
slope, the number in front of X. How about
this one?
Well it'll still work. Seven is the slope.
What's the y-intercept? Well, what's the
number sitting by itself, the constant?
You could have written plus 0.
OK. So you kind of have to imagine that
and the y-intercept will be 0. In this
one you kinda have to imagine a number in
front of X. We know the number sitting by
itself is the y-intercept. What would the
slope be, the number in front of x?
You could write it like this, couldn't
you?
The slope is 0 and if you think about it,
y equals 3, the graph does have a slope
of 0. Y equals 3; it's a flat line isn't
it?
How about the slope and y-intercept of
this equation?
Remember you can't do anything until you
solve for y. Now who's keeping y from
being by himself? The 6x so I'll subtract
6x from both sides which I really can't
do on the right. I just write eight minus
6x. They'll be telling me 2 something
something and on the left side I still
have to get rid of that 2 that's
multiplying the Y. So I'll divide both
sides, everybody on both sides and I'll
get y equals 4 minus 3x. Now it's easy.
The slope is the number in front of X minus
3 and the y-intercept is the other
number 4. Got it?
Let's work backwards.
I want you to give me the equation this
time. Give me the equation of a line with
slope 5 and y-intercept minus 3.
Remember it's always going to be of this
form where the slope is the number in front
of X and the y intercept is the number
sitting by itself, the constant. So
there's your answer:
y equals 5x minus three. Very easy if you
know the form.
Even if I said give me the equation of
slope dadadada and Y heeee and I need to
know this.
the equation will be y equals wococo x
plus bluuuuha. Just fill them in. Get the idea?
How about finding the equation of a line
with slope 2 that contains the point (1,5)?
Well right away I'd take the 2 and put
them into the slope but I still have
to find the b but I do have added
information here. I have a point that
fits. What I can do is make it fit.
Remember the first number is the x and
the second number is the y. So let's put
the 1 in for the x and the 5 in for
the y and make it fit. Well, if five
equals 2 times 1 plus b or 2 plus B, what
is does B have to be? Subtracting 2 from both
sides b will have to be three. And
that's not the answer. The answer is the
equation. So I take what I know to be the
slope which is 2 and what I know to be
the B or y-intercept which is 3 and
I get the equation. Now wait a minute, why
would we even want the equation? Because
once you have the equation you can find
other points of this function if you
would. You can find all kinds of points
once you have that general equation and
that's what we'll use it for.
This may be the equation for converting
amongst temperatures or the equation for
predicting stock prices or once you have
that equation you can use it.
Let's try it again. Make sure you can
follow. To get the equation of a
line with the following clues: a slope of
negative 3 and it contains the point
negative (-2,8). It's got negatives coming
out all over the place. When I use my general
equation, put in the slope that they give
me and now I've got to make that b fit.
Let's put in the first coordinate for X
and the second coordinate for Y
Let's see, negative 3 times negative two
is six so I'll solve for b. Probably do
it in your head. And b is going to have
to be 2 but remember they asked for
the equation. So there's your answer.
We could use this to graph lines.
When I graph the line containing the
y-intercept of two and a slope of minus
1 over 2. Well that has a y-intercept
of two. That's the equation and that's
the y-intercept because it hits the
y-axis at 2 and remember what the
slope means. Between any two points I'm
going to execute the slope of going down
1 and over 2 and find another point. So
really if I had the equation I could
draw the line just that quick.
How cool is that!
That's going to be very useful. Let's try
it again. Let's draw the graph of the
line that has a y-intercept of 1 and a
slope of 3. Right away we can come up
with the equation: y intercept of 1 and a
slope of 3. The three is in front of the
x and the 1 is the constant. Well, I can
put the y-intercept. What does it mean to
be the y-intercept? It means it hits the
y-axis at 1 so the point (0,1) is on this line.
The other thing is that the slope is 3
and 3 as a fraction is three
over one, isn't it? So up three and over 1
and we'll have another point. Once we have
two points
you're there and you could go up three
and over one from the next point also but
you'd still be on that line.
So, you've got the line and the equation
to boot.
Let's try and do two at a time. If you
consider drawing two lines at a time,
three things can happen.
One is that they could look like this
and be what we call parallel, never meet
or they could intersect. Right? These two
lines intersect.
What's the other third possibility?
If you don't guess it is that the two lines
are actually
the same line. That could happen too but
doesn't happen very often. So those are
the three possibilities: parallel means
no points in common, intersect means one
point in common and same line of course as
you can see have infinite number of
points in common.
I'm gonna have to ask you given the
equations which of the three is going on
here? And you should be able to tell me
actually without drawing the picture.
Well,
we know we can't do anything until we
solve for y. I'll solve the first one for y
subtracting 6x. And we've done this one
before and divide everybody by two and
there's that one solved for y. And this
one divide everybody by four and we
have y equals four minus three x versus y
equals 2 minus 3x. Well, we're not even
going draw the picture, I know that the slopes
are the same. Think about two lines that
are leaning the same way. They have the
same slope but they're different.
Now they're different lines because the
y-intercepts are different. Two lines that
have the same slope but different
y-intercepts.
I didn't even have to draw them. I know
they're parallel. That's what it means to
be the same slope. Let's try some more. This
line and this line. We can't do anything
until we solve for y. So let's subtract
6x + divide by 2. You get pretty good at this after
a while.
Let's take a look at this one, I'm going
to subtract 12x and I guess I'll have to
divide everybody by four and this might
stare you're right in the face. Slopes are
the same and the y-intercepts are the same.
You might not have guessed it but those
lines really are the same line. You
couldn't tell anything until you solve
for y, could you. Let's do one more. Let's solve
the first one for y.
You can't do anything until you solve for y.
Who's keeping y from being alone here?
It's the two. That's all; he's the only
troublemaker, Dividing by 2 I get y
equals 3x plus 4 subtracting here and
dividing
I get y equals 2 minus 5x over 2. Now
that looks difficult but if you think
about it
all that matters is that the slopes are
different. As soon as the slopes are
different, 2 lines that have different
slopes considering the lines go on
forever
they have to touch. They may touch in
heaven or hell but they have to touch
and if they touch they intersect.
They intersect in one point so the point of this
is you can do this; you can analyze these
problems without even drawing a picture.
So you need to practice that.
Let's go do the homework.