Since pictures are worth a million
words, and they're going to be especially useful in predicting
outcomes, I'm gonna ask you to take an equation...
a linear equation...
and come up with the picture of it... which would be a line.
Do you remember
when I told you that, if we took all the points
that fit into an equation, or that where the solution to an equation,
here are four of them...
... and plotted them... Do you remember what happens?
let's plot (4, 5)
over 4 and up 5, plot (1 -1)...
over one and down one...
... hmm Now plot (1.5, 0)
I'm gonna go over one-and-a-half squares
and up none... see that... over one-and-a-half squares and up no squares.
and (2,1) ... go over 2 and up one.
Now they're spread out differently but one thing's for sure...
if I did other points... funny thing happens...
They all line up! Every pair
that fits into this equation; that is a solution to this equation...
is on that line. And in fact that line goes on how far?...
forever!
so you can predict the future almost
of this equation which can represent some very useful things in life
with a drawing
of the line. Now if you're trying to find that line...
all you really need is
two points.
Now I'm gonna demonstrate something to you
It doesn't matter which two points I pick.
If I pick to these two (and I didn't even pay attention to the other two)...
I know where the line is wouldn't I? What if I pick these two?
I get the same line. What if I pick these two?
it doesn't matter, I really only need
two points. I'm gonna tell you to do three though.
So here's a kind of problem I'm gonna give you... Graph
this line... Draw the line that represents
this equation.
I know there's an infinite amount a points on the line.
And I know you only need 2 of them.
But as I said I'm gonna ask you to find 3 of them. And I'm gonna ask you to
make a chart. Now you get to pick
any numbers to put in, and I suggest that you put them in for the variable x.
...so let's make them easy numbers
The easiest number I know is zero so I always use 0.
Put in 0 for x and nicely
I can find y. I see y equals
two times 0 which is 0 minus 1.
In this case y is -1 and the the pair (0, -1)
is a solution and I'll graph it... I'll go over none
and down one. Now I need at least one more...
What's the next easiest number you can think of?
I'll put in 1. I put in 1 for x...
two times one is... 2 minus 1...
in this case y comes out as... 1
So I have the pair (1. 1)
I start at the origin and go over 1
and up 1 and there's the point. Now you probably already know where the line is...
but I always tell people to do one more point. And the next
easiest number... use the next easiest number... is two.
2 times 2 is 4...
4 minus 1 is 3... y is 3.
So let's start at the origin... go over 2 and up 3. Now before we do it let's
pray...
that they line up. And I can see...
that they do. The reason I have you do 3 points (and not just 2)...
is, if we do three and they don't line up you can see...
...when you made a mistake.
So always do three points and if they line up...
like this... You know you have the right answer. Now you have, what we still need
to do is... Connect them
and we
know where the line is.
Let's do another one.
I hope you noticed in the last one that it was way easier to find y if we began
by getting y by itself. I'm gonna tell you now... always do that!
Make your chart and begin
by solving for Y FIRST!
You see that...
4x is keeping y from being alone so I'll subtract 4x from both sides.
And I'll get... 8 - 4x
Now, who's keeping y from being alone? 2 is.
so I'll divide EVERYBODY by two.
and I'll get... y equals 4 minus 2x. Now it's going to be much
easier that way, and there's going to be way more reasons in the future for
solving for Y, so get used to (for cripes sake)
ALWAYS solving for y (when graphing)! Now...
We're back to the same old way. Put in the three
easiest numbers I know...
Put in 0. 4 minus two times zero
is... 4
So I have a pair (0, 4)
Go over 0 and up 4. What's the next number I should put in? Let's make life
easy
are put in one for x ...
and four minus two times one is... 2
So in this case, for (1, 2)... I go over one
and up 2. Now I ask you to do 3 points remember so you can check your
errors if you make one...
so we'll put in...
for chripe sake put in easy numbers knucklehead!
we could put in -87 but we could also slap ourselves in the face.
Letss put in an easy number... like two
I'd see 4 minus 2 times 2 ...
or 4 minus 4... we're gonna get 0.
0 can be an answer!
I'm gonna graph the pair (2, 0) by going over 2 and up
none.
I'm very happy about this aren't I... because
they line up. I draw the
line and put on the arrows and I have the graph, if you would, of this linear
equation. Now I got a tough one for you...
See if you can do it. We're going to graph
this equation. Remember how to do it? How do we begin?
always begin by SOLVING FOR Y !
And you're gonna be glad in the future if you did that.
Now let's look at who's keeping y from being alone...
there's a 2x and a 3. Let's begin
by adding 2x ...
to both sides
Now I'm gonna divide everybody by 3. Be careful. We're gonna do it to everybody.
On the left side we just get... y
On the right side we have three over three... which is one...
and two over three which is... two over three.
And it doesn't come out nice. We get a fraction.
But I have a a fix for that. That's why I wanted to do this example.
Usually we've been putting in the three easiest numbers that we know...
0, 1, and 2
We wanna make life easy. And we can put in anything you want...
So instead of using 0, 1, and 2 ...
...let's put in, to make life easier...
put in multiples of the denominator which is in this case is... 3
So we're gonna put in multiples of three and its gonna come out much nicer.
Watch. We're gonna put in 0
That will cancel...
I get a 1. That's easy!
Go over zero and up one. Now let's put in 3. Remember, a multiple of 3...
look the 3's cancel (that's why I put in a 3) ...
and I'm left with one plus two. That's a nice easy
number.
And I'll graph that. Go over three and up three...
Now let's put in... I could put in six also but I chose....
but I chose to put in... -3
and the 3's are gonna cancel but the negative isn't...
and I'm gonna have a -2 plus 1
which is... -1
Can you graph that point? Basically we're gonna go ...
back 3 and down 1 , and son of a gun, they line up !
And all you have to do is draw the
line and put the arrows on it.
So make sure, if you have a fraction
use multiples of the denominator to make life way easier for you.
Tis one is a little bit different but it's still follows the same steps.
Sometimes things are so easy, they're hard.
It still follows our rules... What's your first step?
SOLVE FOR Y ! Get used to that.
Who's keeping y from being alone? 3 .
And he's multiplying
So I'll divide. So in this case... y = 2
As I said, sometimes things are so easy they're hard.
There's no fraction here so I don't need multiples of anybody. I'm gonna put in
the three easiest numbers I know...
(for x) and what are the three easiest numbers?
0,1, and 2 .
The problem here is there isn't an x to put the values in for.
But remember what we're trying to get here... We're trying to get
the y coordinate. So, for the first point...
What is y? What is y? Well we told you right there... y is 2 !
I told you ...it's so easy... it's hard.
y = 2, We didn't even have to put it in. We graph that point...
Go over none and up 2. In the second point guess what y is .
I told you...
y is 2.
Go over 1 and up 2. For the third point what is y?
You got it... y is 2 !
and nicely if we graph (2, 2); go over 2 and up 2 ...
You're gonna feel a lot better. You are correct sir.
They line up. This one is called a "constant" function. Tt's "constantly" equal to 2.
but, son-of-a-gun, it is a linear (forms a line) function.
It forms a line.
Every time... y is 2.
Now the "solve for y" method is gonna work the vast majority of the times but...
there is one situation when it won't work and I'd like you to see that.
In this case...
What do I always ask you to begin with?
SOLVE FOR y ! ...but they're isn't any y.
So what do we do now?
you don't have to be a brain surgeon to know... We have to solve for x.
We don't have any other choice Bub.
So we'll solve for x. But normally I try to solve for y.
There isn't a y so we'll solve for x. Who's keeping the x from being alone?
the two. He's multiplying so...
I'll divide by two.
now get x = -1. Now, as I said, this one's a little bit weird but not difficult.
What should we put in for x? Be careful here.
What are we gonna put in for x in our three points?
Read the equation. x has gotta be -1 they said it was -1.
they did not say anything about y.
Since they didn't say anything about y, y can be anything.
and I get to pick y. I pick the
easiest number; 0. I graph (-1, 0)...
...back 1 and up none, and it goes there.
Now you have to pick the first point's y. what are you gonna pick for the second
point's y value?
Pick anything you want...
...the next easiest number... I pick 1. Go back one...
and up one. Hey, thery're starting line up here
Now a third point. No fraction so let's use 2.
Go back one and up 2.
I'm feeling pretty good. They line up.
I draw it and ... It's OK.
All right, go practice. DO YOUR HOMEWORK!
okay we'll go practice do your homework