What we're going to do in this video is graph some lines
using the x- and y-intercepts of the line.
So first of all, a line is defined by two points.
If you find two points-- if you're able to plot two points
on a line, then you can connect them and keep going in
both directions.
And you will have drawn the line.
What I'm going to do in this video-- the two points I'm
going to pick are the intercepts of the lines.
There's two of them.
There's the x-intercept and then there's the y-intercept.
These are the points where the lines
intersect the x- and y-axes.
And just to make it clear, this right here is the x-axis
on my graph paper.
That right there is my x-axis.
It just keeps going.
That's my x-axis.
This right here, in the middle, going up
and down, is my y-axis.
That right there is my y-axis.
I don't think you can see what I just wrote.
That is the y-axis right there.
So let me show you what I mean by x- and y-intercepts.
So let's do part (a).
I'll do it in a different color, not the same color as
my intercepts.
Part (a), we have the equation y is equal to 2x plus 3.
So first of all, let's think about what happens when x is
equal to 0.
When x is equal to 0, what is y equal to?
y is going to be equal to two times 0 plus 3.
Well, 2 times 0, that's just 0.
So y is equal to 0 plus 3 or y is equal to 3.
So the point x is equal to 0, y is equal to 3, that
satisfies this equation.
Or it's on the line defined by this equation.
So the point 0, 3.
Let me plot that.
The point 0, 3.
Now notice, this point-- this is when x is equal to 0, this
sits on our y-axis.
We call this right here a y-intercept.
It is when x is equal to 0.
But it's called a y-intercept because it sits on the y-axis.
This is where the line intercepts the y-axis.
Now let's do the same for the x-axis.
Let's set y equal to 0.
So if y is equal to 0, we have 0 is equal to 2x plus 3.
We could subtract 3 from both sides of that equation.
And you get negative 3 is equal to 2x.
We can divide both sides by 2.
And you get negative 3/2 is equal to x.
And negative 3/2, that's the same thing as negative 1 1/2.
So we know that the point negative 3/2, 0, right?
The second scenario, y is equal to 0.
We know that this set of values of x and y also satisfy
this equation.
Or that this point also sits on the line.
So negative 3/2-- remember that's the same thing as
negative 1 1/2.
That's right there. y is 0.
So that's that second point we just figured out.
When y is 0, we're dealing with our x-intercept.
We're intercepting the x-axis.
That is the x-intercept right there.
If I wanted to draw the line, it'll look like this.
Very roughly, I'll connect those points and then just
keep going.
Keep going in both directions.
Keep going in both directions forever and then I will have
drawn my line.
Let's do one more of these.
I don't want to make it too messy over here.
Let me clear all this stuff out of the way.
Let me clear this out of the way as well.
You could always pause it, if you want to gaze at it-- oh I
shouldn't have done that, I got rid of my-- well you can
see still the axes.
I'll redraw the axes.
I'll redraw it right there.
Good enough.
That's one, my x-axis.
That is my y-axis right over there.
Let me do part (b). (b) looks like an
interesting one right there.
(b) has 6 times x minus 1 is equal to 2 times y plus 3.
So let's look at the situation when x is equal to zero.
When x is equal to 0, that equation up there becomes 6
times negative 1.
Right?
Because this is 0.
So 6 times negative 1 is negative 6 is equal to 2-- let
me distribute this-- 2y plus 6.
I multiplied 2 times the y and times the 3.
And then we can subtract 6 from both sides, so that you
get negative 12.
Negative 12 is equal to 2y.
I subtracted 6 from both sides to essentially move this 6 on
to the left-hand side.
You subtract 6 here, this disappears.
You subtract 6 from here, you get negative 12.
Divide both sides by 2, you get negative 6 is equal to y.
So our first point that we know is on the line is a point
x is equal to 0, y is equal to negative 6.
That is our y-intercept.
Now let's see what happens when y is equal to 0.
Go back to the original equation.
We have-- let me write that a little bit different-- when y
is equal to 0, we have 6 times x minus 1.
Let me just distribute that.
So 6x minus 6 is equal to 2 times y.
Well, y is just going to be 0.
So it's going to be 2 times 3.
This whole thing is going to be 3.
It's going to be equal to 6.
So let's subtract 6 from both sides-- or let's add 6 to both
sides of this equation.
I want to get rid of that right there.
So I need to add 6 to make that 0.
So we get 6x is equal to-- you add 6 to the right-hand side,
you get 12.
And you get x-- divide both sides by 6, you get
x is equal to 2.
So our other intercept is x is equal to 2, y is equal to 0.
So we have x is equal to 2, y is equal to 0.
Right there.
That is our x-intercept.
And then our actual line, we just connect the dots.
It will look something like that.
Obviously, you have trouble drawing a straight line, but I
think you got the idea.
Two points define a line.
Let me do one more of these.
I could probably do it up here, in this
real estate up here.
Let's do part d.
You can do part c if you want extra practice.
So part d, we have x plus y is equal to 8.
This is actually very, very straightforward.
When x is equal to 0, what is y equal to?
If this is 0, all we have left is y is equal to 8.
And then when y is 0, what is x?
Well, you can just almost cover this up.
Then you just get x plus 0 is equal to 8.
x is equal to 8.
So that was pretty straightforward right there.
So we have one point, 0, 8.
So 0, 8.
That is the y-intercept.
And then we have the point 8, 0.
8, 0.
That is the x-intercept.
And then we can connect the dots.
It looks something like that.
Now let's do this problem down here.
At a local grocery store, strawberries cost $3.00 per
pound and bananas cost $1.00 per pound.
If I have $10 to spend between strawberries and bananas, draw
a graph to show the combination of each I can buy
and spend exactly $10.
Let me draw my axes.
Make sure that these are nice and filled in.
So that remember is my horizontal access.
But just for fun, instead of calling it the x-axis, I'm
going to call it the strawberry-axis.
Actually let me call it s, the s-axis, where s is for
strawberries.
Let s equal the number of strawberries.
I think you know where I'm going.
b will be number of bananas.
Let me do my b-axis.
I'm going to plot the number bananas on the vertical axis.
So this right there, that is the b-axis.
b for bananas.
Strawberries cost $3.00.
OK, this isn't the number of strawberries.
This is the number of pounds.
Let me clear that up.
This is pounds of strawberries.
We're not going by the number of strawberries.
We're going by pounds of strawberries.
This is pounds of bananas.
All right.
Strawberries cost $3.00 per pound, bananas
cost $1.00 per pound.
So how much am I going to spend?
I'm going to spend $3.00 times the number of strawberries
because they're $3.00.
All right, I'm going to spend $3.00 times the number pounds
of strawberries because it's $3.00 per pound plus $1.00
times the number of pounds of bananas.
And they say I have $10.00 to spend on both.
So that's going to be equal to $10.
So 3 times the pounds of strawberries plus the pounds
of bananas are going to be equal to 10.
1b, that's the same thing as b, so I can rewrite this as 3s
plus b is equal to 10.
Now let's plot this.
Let's look at the situation where I get no strawberries.
So my pounds of strawberries are 0.
What does this equation become?
It becomes 3 times 0 plus b is equal to 10.
That's just 0.
So b will be equal to 10.
So I would have the point 0 pounds of strawberries, I
could get 10 pounds of bananas.
So if I get 0 pounds of strawberries, I can get 10
pounds, right there, of bananas.
Now what about the other scenario?
What about the scenario where I get no bananas?
I get 0 pounds of bananas.
Now let's substitute back here.
We have 3 times my pounds of strawberries plus 0 pounds of
bananas will equal 10.
That's just a 0.
Divide both sides by 3.
I could-- if I get no bananas-- I can get 10/3
pounds of strawberries.
Or this is equal to 3 1/3.
So it would have the point 3 1/3 pounds of strawberries, 0
pounds of bananas.
So 3 1/3 pounds of
strawberries, 0 pounds of bananas.
And then we can connect the line.
This whole line I'm going to draw, I am just going to draw
it in the first quadrant.
Because I can't have negative bananas.
I'm not going to sell pounds of bananas.
And I can't have negative strawberries.
We're not talking about selling strawberries.
So I can only buy these things.
This is the amount that I'm buying.
So let me to connect the dots right there.
This is neat because this line shows all of the possible
combinations of pounds of strawberries and bananas.
For example, this point right here-- I don't know if I'm
drawing it exactly-- it looks like if I get about 5 pounds
of bananas, I can get a little under 2 pounds of
strawberries.
That's what that tells me.
If I get 3 pounds of strawberries, if I get exactly
3 pounds of strawberries, I can get 1 pound of bananas.
Every point here represents a combination that
I can get for $10.