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All right.
In this video, I want to show you how to graph a
equation of a line.
And in this case, the equation is 2x minus 5y equals 10.
Now, perhaps you might have learned in a previous course
or some other way that we could make up a table, an x
and a y table, to graph this thing.
We could make up some x and y table and plug in some numbers
for x or plug in some numbers for y and plot some points.
But I'm not going to do that this time.
I'm going to show you a different way to graph this.
So I'm going to slide this aside for a second.
Let me show you a different way to graph an equation.
First, let's start off with what we were given.
And I'm going to take this equation and turn it into this
form called y equals mx plus b.
Maybe you've seen that form before.
This is a really popular form of an equation of a line.
In fact, it's called the slope-intercept form because
this m, this coefficient of the x, is the slope.
And this b over here, this number that's going to be
sitting all by itself, is called the y-intercept.
So what we need to do to this equation, 2x minus 5y equals
10, is to make it look like this one over here.
We want y all by itself and everybody else on the other
side of the equation.
So the first thing we're going to do is we're going to move
this 2x over to the other side of the equation.
To do that, we're simply going to subtract
2x from both sides.
Have you ever heard the phrase change sides, change signs?
So in this case, if I'm going to bring that positive 2x to
the other side of the equals sign, it
becomes a negative 2x.
OK, well I have to do one more thing to get my y
completely by itself.
I have to get rid of that negative 5.
And since negative 5 is being multiplied by y, we're going
to divide every term by a negative 5.
Divide every term by negative 5.
That now leaves us with a coefficient of 1 in
front of the y.
And since I have a negative 2 divided by negative 5-- hey, a
negative divided by a negative,
that's just a positive.
So I'll just write it as a positive 2/5.
And obviously, 10 divided by negative 5 is a negative 2.
OK, so now this equation looks like what I was after, this
slope-intercept form.
And I've got all the players that I need.
I hope you see--
if you see it in this form--
that this coefficient of x, this thing in front of the x,
is my slope.
This guy here is my slope, 2/5.
And this negative 2 sitting here all by itself at the very
end of my equation is what we call a y-intercept.
So to graph an equation like this, we don't have to plot
any coordinates or anything like that.
What we're going to do instead--
let's see if I can keep this in there.
Actually, maybe what I'll do is I'll just rewrite the
equation this way.
So what we're going to do instead then is we're going to
start off by plotting the y-intercept.
That's the first thing we're going to do.
It's always the first thing we're going to do when you use
this form here.
So on the y-axis, I need to find negative 2.
So I've got my graph paper here.
I'm just going to go down, negative 1, negative 2, and
I'm going to put a dot right there.
Maybe I should do this in red so you could see it in
contrasting colors here.
So I'm going to put a dot here at negative 2.
And the reason I put that there, again,
is I did this first.
I did the y-intercept first, OK?
Then from there, we're going to do the slope.
Now, slope is always rise over run.
So from this point, not from the origin, but from that
y-intercept, I'm going to go up 2.
I'm going to go up 2.
That's 1, 2, up.
And then to the right 5--
1 2, 3, 4, 5.
And then put a second dot.
OK, so that is called--
this little shortcut here is using the slope-intercept form
of an equation of a line to graph.
Our line looks like-- and I'll just have to
connect these dots.
That's my last step here.
Connect these dots up, and our line looks
something like that.
I like to draw nice long lines.
No wimpy small lines, OK.
So there it is.
Without even doing a table at all, not even using that
method at all, all I did was I isolated the y.
And now that I've got my y completely by itself, I have
an equation where I can pick off the y-intercept.
That's the first thing I'm going to find.
And then from there, I'm going to do the slope.
Well, let's say, for example, I had-- let's make up one more
of these-- let's say for example I had an equation that
looked like this.
This is just a completely separate question now.
What if I had something like negative 3/4x plus 7,
something like that?
What would that look like?
Well, let me show you how that looks.
And it's slightly different because it's got a negative
slope this time.
Well, again, the first thing we're going to do is we're
going to tackle that y-intercept.
We got to go find that first.
So I'm just going to go up 1, 2, 3, 4, 5, 6,
7, and put a dot.
So the y-intercept is the first thing we do.
And again, it's a y-intercept, not an x-intercept.
Notice I'm not finding anything on the x-axis first.
That is the y-intercept.
OK, so from there, what I like to do when I tackle these
things is since this is negative--
this is a negative slope--
I'm going to go down 3--
1, 2, 3--
and to the right 4--
1, 2, 3, 4-- and then put a dot.
So I went down 3.
From here, I went down 3, to the right 4, put a dot.
And if I connect them up, I've got the line that corresponds.
I like to put arrows on the end because these technically
are lines, and lines go on and on forever.
So this equation here is this line here.
That top equation here was this line here.
So negative slopes, when you have a negative slope, they go
down as you're looking at it from left to right.
And positive slopes like this one up here go up as you're
looking at it from left to right.
Now, one more thing I want to show you.
What if we play the game slightly backwards?
What if the game is played this way?
What if I give you the graph, and I ask
you what's the equation?
We could do that too.
My math lab does that sometimes to you, or the
homework might do this to you.
So what if I give you the picture, and I ask you to find
the equation of the line in slope-intercept form?
Well, slope-intercept form, again, always starts off like
this, where y is all by itself.
We have this thing in front of the x called the slope, and we
have this thing over here all by itself, this number all by
itself, a constant, which is where it crosses the y-axis.
So let's start with that.
Where does this line, this red line that I have here, where
does it cross the y-axis?
Well, I hope you see it crosses the y-axis right here.
This thing right here is called the y-intercept.
And we'll see in my case it's 1, 2, 3, 4, 5.
That's a positive 5 sitting right there.
So here's what I know so far.
I know that this thing, this number that's sitting here all
by itself, is a positive 5.
I know that for sure.
If it crossed down here somewhere, it would be a
negative number.
It would be minus something.
But since it's crossing right here, it's a positive 5.
So I'm just going to put a plus 5.
So the last thing, then, is I need to
know what is the slope.
Now remember what I said a moment ago.
When you see a line going down from left to right--
I'm reading the line this--
I'm looking at it panning across the page this way.
If when you see the line going down like this, down, I know
it's got to be a negative slope.
I know it's a negative something.
OK, so automatically I can put the minus sign
in there right now.
Minus sign in there.
I know it's going to be negative something.
I was also given this other point over here.
So what I'm going to do is I'm going to go--
from here, I'm going to go down 1, 2, 3, 4, 5, 6, 7, 6,
7, and then over to the right 1.
So I'm going to go down 7, to the right 1.
Hey, that's a 7 on top and a 1 on the bottom.
Because remember that slope is always rise over run.
It's always the up and down stuff over the left and
right stuff, OK.
So from here to here, you can go down 7 and then
to the right 1.
And that takes you to this other point
that we were given.
So I'm going to finish this off.
I guess I really don't need the 1 on the bottom.
I'm going to finish this off and say, OK, the equation of
this line is y equals negative 7x plus 5.
There's the equation of that line.
And if there is any other denominator other than a 1, I
would include it.
Maybe if we want to the right 2 or 3 or 4 or something like
that, I would include that instead.
But since it's over 1, I don't need to include it.
Hope that helps.