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>> Okay, we're going to go over some more examples.
Graphing lines.
There's two methods we've gone over so far.
Using a T chart where you pick a value of X and put it in the equation
and solve for Y to get an ordered pair.
And you find a few ordered pairs and graph the line.
And the other method we did was the X and Y intercept method,
where you find the X intercept and the Y method.
The X and Y intercept method works for most lines, but not if it's going through the origin,
or if it's a vertical or horizontal line.
So let's start with an example.
5 X minus 3 Y equals 15.
Now since this is not solve for Y, let's try the X and Y intercept method.
Okay, so to get the X intercept remember you are going to put in zero for Y. Right?
So your ordered pair, you're halfway there, put in 0 for Y into the equation.
So we have 5 X minus 3 times 0 equals 15.
Now you don't necessarily have to write all this out, you might be able to look at it and realize
when you put in 0 for Y you'll simply get 5 X equals 15.
And then if you divide both sides by 5 you get X equals 3.
So 3, 0, is the X intercept.
So let's get the Y intercept now.
And the Y intercept, remember, we put 0 in for X. Because everywhere
on the Y axis you're not moving left or right at all.
So if we're putting in 0 for X we have 5 times 0 minus 3 Y equals 15.
Which says negative 3 Y equals 15.
And if you divide by negative 3, be careful, we get negative 5.
So we've got 0, negative 5.
So here's a way of graphing the line by using the X and Y intercepts.
We find each of them, we'll plot the X and Y intercept, and then graph it.
All right, so you get out graph paper and let's graph this.
We have 3, 0, we'll be right here.
And 0 negative 5 will be right here.
And then we draw carefully the line going through those two points.
And as long as we didn't make a mistake we've got the line graphed for 5 X minus 3 Y equals 0.
So we use the X Y intercept method.
Let's do this same problem using a T chart.
So I'm going to go ahead and leave the graph we got by using the X and Y intercept method.
And to use a T chart let's solve for Y. So we have negative 3 Y equals negative 5 X plus 15.
Divide both sides by negative 3.
And we have 5/3 X minus 15 -- minus 5.
So we've got Y equals 5/3 X minus 5, and now we would plug in some numbers for X,
and then compute 5/3 X minus 5 to get a value of Y. Now
since we're doing 5/3 X let's pick values of X that are multiples of 3.
So negative 3, 0, and 3 might be a good choice.
So we have 5/3 times negative 3 minus 5.
All right, so what do we get there.
3s cancel.
5 times negative 1, so negative 5 minus 5 is negative 10.
And then we're going to plug in 0.
Let's see, could do that in your head.
If you put in 0 up here in the equation you're going to get Y equals negative 5.
So when you put in 0 you're going to get negative 5.
I'm not going to show that computation.
And then 5/3 times 3 minus 5.
The 3s will cancel, and 5 minus 5 is 0.
So let's do this with some green marker.
I'm going to plot these points.
Well, negative 3, negative 10 -- ah, that's way off the chart, it's somewhere down here.
I could estimate it.
Let's try that.
One, two, three -- and then this goes -- there's negative 6, right?
So that's negative 8, 9, 10.
So it looks like it's about right here.
Yeah, it looks like it's going to be on that line.
I'm approximating, of course.
Let's do 0 negative 5.
Hey, same thing as I had when I did it the other way, which was the X and Y intercept.
And 3, 0. Hmm, that just also happened to be the X intercept.
I don't know if I -- the 0 negative 5 was the Y intercept.
Sorry. And so you can see that this is going to be same line.
I'm using a T chart coming up with the same graph.
Now which do you think is easier?
I think it was easier to do the X and Y intercepts, because it is simple to plug
in a value for -- plug in 0 for X and solve for Y, and then plug in 0 for Y and solve for X.
So I would use the X and Y intercept method if your original equation here is not solve
for Y. Usually it's easier to use the X and Y intercept method.
Keep in mind, though, you might get a fraction for the X or Y intercept.
But that's okay, fractions are numbers too.
Here's another problem.
3 X plus 2 Y equals 0.
So this is not solve for Y. So let's try using the X and Y intercept method.
So we try getting the X intercept by putting in 0 for Y, remember?
So we put in 0 for Y and hopefully you see that's going
to give you 3 X plus 0, or 3 X equals 0.
Okay? So if you divide by 3, we're going to get X equals 0.
So I got 0, 0.
You know what, if 0, 0, is one of the points on the line the origin is on the line,
and you're not going to get a different X and Y intercept.
You've got the same X intercept as you do for the Y intercept.
So if you plugged in 0 for the other value, so in other words if I was trying
to get the Y intercept look what would happen.
You would put in 0 for X and gotten 2 Y equals 0.
And when you divide by 2 you get the same points.
So there's a problem here.
I wasn't able to get two different points, so how can I draw a line if I only have one point.
So in this particular example notice the constant.
I've got 0 over here.
If 0, 0, is a point on the line, the X and Y intercept method is not going to work.
So you know what, we're going to have to use a T chart.
That's one of the methods we could use, I better say that.
It's just that you do not have to use -- I mean, the X and Y intercept method will not work.
All right, so I at least have the 0, 0.
So there's nothing wrong with leaving that as one of my ordered pairs.
But let's go ahead and try solving for Y. So if I solve for Y, I'm going to subtract 3 X
from both sides and then divide by 2 so that Y equals negative 3/2 X. So let's plug
in some values for X into negative 3/2 X.
And find some values of Y. Now remember we already know 0, 0, is on the line.
So I don't need to do that again.
Let's just pick two more points.
How about something where 2 is going to cancel for X. So how
about negative 2 and then positive 2.
So we've got negative 3/2 times a negative 2.
And then we'll do negative 3/2 times a positive 2.
Right, so for the first one the 2s cancel, and negatives,
negative times negative is going to be positive.
So my answer is going to be 3.
When I put in negative 3/2 times 2, the 2s cancel, I get negative 3.
This gives me two other ordered pairs, negative 2, 3, and 2, negative 3.
And now I've got -- well, as long as you have two points it's going to work.
But it's even better if you can get three points on the graph.
So I've got three points on the graph because I already figured out 0, 0, was on it.
And there's my line.
So here's the deal, if it's solve for Y,
it's easier to use a T chart than the X and Y intercepts.
So if I had given you the equation of Y equals negative 3/2,
this would probably be the simplest way to do it of these two methods.
Or if I gave you Y equals 3 X plus 1, or anything where it was solve for Y.
If it's not solve for Y try to use the X and Y intercept method.
And if it doesn't work because it goes through 0, 0, then you can solve for Y and proceed.
So let's see.
Here's our four equations.
And let's say you had to graph them.
Which one do you think would be easier to use to graph these equations.
Think about that.
The T chart where you just find some ordered pairs or the X and Y intercept method.
Go ahead and figure that out.
And put it on pause, and then come back.
All right, the first one.
It's solve for Y. So you know what, T chart.
May as well.
Just plug in some -- any number you want for X and solve for Y.
And you've got -- get a few ordered pairs.
Second one.
It's not solve for Y. So you would probably use the X and Y intercept method.
It's pretty easy.
You plug in 0 for X, find the Y value.
That will give you the Y intercept.
And then plug in 0 for Y, and get the X value.
That will get the X intercept.
How about this one, X minus 3 Y equals 0.
At first glance you might think, well, I would try to use the X and Y intercept method.
But if you try that you'll find out that 0, 0, is an intercept.
So X and Y intercept method won't work here and you'll have to end up using the T chart.
And the last one is solve for Y, so you may as well use the T chart.
But this is just what I think might be easier.
The X and Y intercept method will work on all of these except the third one.