In this lesson, we're going to graph some straight lines;
and the first equation we're given we have some partially filled in ordered pairs.
We want to graph x + y = 3 and I'm going to do that by filling in these ordered pairs.
So, when x = 0 right here, I have 0 + y = 3, y must be equal to 3.
When x = 2, I have 2 + y = 3, y must be equal to 1.
And when y = -1, let's see I have x + (-1) = 3. If I add one to both sides,
I end up with x = 4. So, if I graph these three ordered pairs, (0, 3), x = 0, y = 3, right there.
(2, 1), so x = 2, y = 1. And then, (4, -1), so 1, 2, 3, 4, -1.
I get those three points, they line up in a straight line, so I'll draw a line through them,
and that is a picture of all the solutions to the equation x + y = 3. So, there's my line and x + y = 3.
It's a straight line all the ordered pairs. Any ordered pair that's on this line, any point on this line,
has coordinates or an ordered pair associated with it that's a solution to this equation.
And, any ordered pair that's a solution to this equation has a graph that lies on this line.
Let's try another one. How about y = -(2/3)x + 1. So here, I have x = -3.
Let's see what that looks like. (-2/3)(-3) + 1. A negative times a negative is positive.
3 divides into 3 once, times 2 is 2, so I have 2 + 1. So, y = 3. So, when x = -3, y = 3.
When x = 0, I have (-2/3)(0) which is 0 + 1 = 1, and when x = 3, I'll have (2/3)(3),
which is 2 and it's negative. So, I'll have a -2 + 1, and I'll end up with y = -1.
Let's graph these three ordered pairs. x = -3, y = 3. So that's that point right there.
(0, 1), x = 0 and y = 1. (3, -1). So, x = 3; y = -1.
So, there again I have three points. They line up in a straight line. I'll draw a line between them,
and that will be the graph on that equation. So, this line is the graph of this equation and any ordered pair that
satisfies this equation has a graph that lies on this line.
And, any point that lies on this line has coordinates that satisfy this equation.
Let's try another one in which we choose the ordered pairs ourselves. So, here I have -2x + y = 1.
Let's let x = 0. Then, I have -2(0) = 0, + y = 1, so y is equal to 1.
Let's let x = 2 and then I'm going to have, let's see, -2(2) + y = 1.
That's -4 + y = 1. Add positive 4 to both sides. y will be equal to 5. So, I'll let y = 5,
or, y ends up being 5 right there, and let's just get one more point for insurance. How about a negative 1 for x;
and I have a -2(-1), that's a positive 2 + y = 1. y must be equal to -1 in that case.
So, I want to graph these three ordered pairs. (0, 1), that's that point. (2, 5), so x = 2, y = 5.
(-1, -1). That point right there. So, you can see that those three points lie in a straight line.
I'll draw a line between them and that will be the graph of that equation.
So, that straight line is a picture of all the solutions to this equation right here.
The last two we want to do is x = -2, y = 2. So, let's take a look at those.
x = -2. That means that y can be anything, so x = -2, y can be 5. x = -2, y can be 0.
x = -2, y can be negative 2. It doesn't matter what y is as long as x = -2 in this equation.
So, -2 for x, y = 5. x = -2, y = 0. So, (-2, 0)
x = -2, y = -2, so (-2, -2). So, there I have those three points.
You can see they lie in a vertical line. I'll draw a line between them that contains all those points,
and that is a picture of all the solutions to the equation x = -2.
One last problem: y = 2. So, I end up with three ordered pairs. It doesn't matter what x is as long as y = 2.
So, x can be 0, y = 2, x can be 3, y = 2. x can be -1 as long as y = 2. So, (0, 2), that's x = 0, y = 2, that's that point.
x = 3, y = 2. (-1, 2)
You can see that those three points lie in a horizontal line. Draw a line between them,
and that's the graph of the y = 2. So, every equation that has this form:
y equal some number is going to have a graph that is a horizontal line.
Every equation that has this form x equal a number is going to have a graph that is a vertical line like that.
Quick look at graphing straight lines.