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So you may or may not already know that any linear equation

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can be written in the form y is equal to mx plus b.

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Where m is the slope of the line.

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The same slope that we've been dealing with

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the last few videos.

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The rise over run of the line.

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Or the inclination of the line.

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And b is the y-intercept.

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I think it's pretty easy to verify that b is a

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y-intercept.

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The way you verify that is you substitute x is equal to 0.

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If you get x is equal to 0-- remember x is equal to 0, that

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means that's where we're going to intercept at the y-axis.

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If x is equal to 0, this equation becomes y is equal to

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m times 0 plus b.

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m times 0 is just going to be 0.

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I don't care what m is.

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So then y is going to be equal to b.

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So the point 0, b is going to be on that line.

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The line will intercept the y-axis at the point

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y is equal to b.

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We'll see that with actual numbers in

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the next few videos.

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Just to verify for you that m is really the slope, let's

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just try some numbers out.

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We know the point 0, b is on the line.

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What happens when x is equal to 1?

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You get y is equal to m times 1.

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Or it's equal to m plus b.

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So we also know that the point 1, m plus b

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is also on the line.

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Right?

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This is just the y value.

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So what's the slope between that point and that point?

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Let's take this as the end point, so you have m plus b,

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our change in y, m plus b minus b over our change in x,

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over 1 minus 0.

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This is our change in y over change in x.

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We're using two points.

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That's our end point.

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That's our starting point.

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So if you simplify this, b minus b is 0.

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1 minus 0 is 1.

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So you get m/1, or you get it's equal to m.

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So hopefully you're satisfied and hopefully I didn't confuse

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you by stating it in the abstract with all of these

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variables here.

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But this is definitely going to be the slope and this is

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definitely going to be the y-intercept.

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Now given that, what I want to do in this exercise is look at

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these graphs and then use the already drawn graphs to figure

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out the equation.

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So we're going to look at these, figure out the slopes,

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figure out the y-intercepts and then know the equation.

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So let's do this line A first. So what is A's slope?

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Let's start at some arbitrary point.

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Let's start right over there.

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We want to get even numbers.

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If we run one, two, three.

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So if delta x is equal to 3.

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Right?

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One, two, three.

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Our delta y-- and I'm just doing it because I want to hit

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an even number here-- our delta y is equal to-- we go

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down by 2-- it's equal to negative 2.

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So for A, change in y for change in x.

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When our change in x is 3, our change in y is negative 2.

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So our slope is negative 2/3.

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When we go over by 3, we're going to go down by 2.

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Or if we go over by 1, we're going to go down by 2/3.

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You can't exactly see it there, but you definitely see

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it when you go over by 3.

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So that's our slope.

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We've essentially done half of that problem.

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Now we have to figure out the y-intercept.

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So that right there is our m.

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Now what is our b?

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Our y-intercept.

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Well where does this intersect the y-axis?

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Well we already said the slope is 2/3.

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So this is the point y is equal to 2.

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When we go over by 1 to the right, we would have

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gone down by 2/3.

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So this right here must be the point 1 1/3.

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Or another way to say it, we could say it's 4/3.

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That's the point y is equal to 4/3.

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Right there.

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A little bit more than 1.

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About 1 1/3.

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So we could say b is equal to 4/3.

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So we'll know that the equation is y is equal to m,

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negative 2/3, x plus b, plus 4/3.

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That's equation A.

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Let's do equation B.

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Hopefully we won't have to deal with as

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many fractions here.

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Equation B.

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Let's figure out its slope first. Let's start at some

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reasonable point.

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We could start at that point.

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Let me do it right here.

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B.

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Equation B.

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When our delta x is equal to-- let me write it

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this way, delta x.

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So our delta x could be 1.

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When we move over 1 to the right, what happens

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to our delta y?

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We go up by 3.

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delta x.

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delta y.

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Our change in y is 3.

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So delta y over delta x, When we go to the right, our

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change in x is 1.

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Our change in y is positive 3.

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So our slope is equal to 3.

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What is our y-intercept?

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Well, when x is equal to 0, y is equal to 1.

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So b is equal to 1.

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So this was a lot easier.

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Here the equation is y is equal to 3x plus 1.

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Let's do that last line there.

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Line C Let's do the y-intercept first. You see

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immediately the y-intercept-- when x is equal to 0, y is

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negative 2.

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So b is equal to negative 2.

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And then what is the slope?

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m is equal to change in y over change in x.

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Let's start at that y-intercept.

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If we go over to the right by one, two, three, four.

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So our change in x is equal to 4.

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What is our change in y?

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Our change in y is positive 2.

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So change in y is 2 when change in x is 4.

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So the slope is equal to 1/2, 2/4.

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So the equation here is y is equal to 1/2 x, that's our

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slope, minus 2.

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And we're done.

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Now let's go the other way.

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Let's look at some equations of lines knowing that this is

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the slope and this is the y-intercept-- that's the m,

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that's the b-- and actually graph them.

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Let's do this first line.

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I already started circling it in orange.

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The y-intercept is 5.

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When x is equal to 0, y is equal to 5.

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You can verify that on the equation.

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So when x is equal to 0, y is equal to one, two, three,

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four, five.

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That's the y-intercept and the slope is 2.

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That means when I move 1 in the x-direction, I move up 2

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in the y-direction.

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If I move 1 in the x-direction, I move up 2 in

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the y-direction.

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If I move 1 in the x-direction, I move up 2 in

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the y-direction.

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If I move back 1 in the x-direction, I move down 2 in

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the y-direction.

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If I move back 1 in the x-direction, I move down 2 in

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the y-direction.

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I keep doing that.

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So this line is going to look-- I can't draw lines too

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neatly, but this is going to be my best shot.

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It's going to look something like that.

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It'll just keep going on, on and on and on.

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So that's our first line.

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I can just keep going down like that.

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Let's do this second line.

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y is equal to negative 0.2x plus 7.

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Let me write that. y is equal to negative 0.2x plus 7.

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It's always easier to think in fractions.

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So 0.2 is the same thing as 1/5.

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We could write y is equal to negative 1/5 x plus 7.

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We know it's y-intercept at 7.

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So it's one, two, three, four, five, six.

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That's our y-intercept when x is equal to 0.

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This tells us that for every 5 we move to the right,

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we move down 1.

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We can view this as negative 1/5.

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The delta y over delta x is equal to negative 1/5.

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For every 5 we move to the right, we move down 1.

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So every 5.

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One, two, three, four, five.

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We moved 5 to the right.

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That means we must move down 1.

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We move 5 to the right.

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One, two, three, four, five.

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We must move down 1.

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If you go backwards, if you move 5 backwards-- instead of

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00:08:44,740 --> 00:08:47,259
this, if you view this as 1 over negative 5.

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00:08:47,259 --> 00:08:49,450
These are obviously equivalent numbers.

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00:08:49,450 --> 00:08:52,060
If you go back 5-- that's negative 5.

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00:08:52,059 --> 00:08:54,500
One, two, three, four, five.

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00:08:54,500 --> 00:08:56,804
Then you move up 1.

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00:08:56,804 --> 00:09:00,049
If you go back 5-- one, two, three, four,

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00:09:00,049 --> 00:09:01,829
five-- you move up 1.

202
00:09:01,830 --> 00:09:04,110
So the line is going to look like this.

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00:09:04,110 --> 00:09:07,000
I have to just connect the dots.

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I think you get the idea.

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00:09:08,019 --> 00:09:09,169
I just have to connect those dots.

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00:09:09,169 --> 00:09:11,169
I could've drawn it a little bit straighter.

207
00:09:11,169 --> 00:09:13,839
Now let's do this one, y is equal to negative x.

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00:09:13,840 --> 00:09:16,480


209
00:09:16,480 --> 00:09:17,800
Where's the b term?

210
00:09:17,799 --> 00:09:18,939
I don't see any b term.

211
00:09:18,940 --> 00:09:22,490
You remember we're saying y is equal to mx plus b.

212
00:09:22,490 --> 00:09:23,669
Where is the b?

213
00:09:23,669 --> 00:09:24,779
Well, the b is 0.

214
00:09:24,779 --> 00:09:26,769
You could view this as plus 0.

215
00:09:26,769 --> 00:09:28,289
Here is b is 0.

216
00:09:28,289 --> 00:09:31,709
When x is 0, y is 0.

217
00:09:31,710 --> 00:09:35,080
That's our y-intercept, right there at the origin.

218
00:09:35,080 --> 00:09:37,490
And then the slope-- once again you see a negative sign.

219
00:09:37,490 --> 00:09:40,710
You could view that as negative 1x plus 0.

220
00:09:40,710 --> 00:09:42,200
So slope is negative 1.

221
00:09:42,200 --> 00:09:45,570
When you move to the right by 1, when change in x is 1,

222
00:09:45,570 --> 00:09:47,690
change in y is negative 1.

223
00:09:47,690 --> 00:09:52,320
When you move up by 1 in x, you go down by 1 in y.

224
00:09:52,320 --> 00:09:55,580
Or if you go down by 1 in x, you're going to go up by 1 in

225
00:09:55,580 --> 00:09:57,660
y. x and y are going to have opposite signs.

226
00:09:57,659 --> 00:09:58,850
They go in opposite directions.

227
00:09:58,850 --> 00:10:03,955
So the line is going to look like that.

228
00:10:03,955 --> 00:10:06,550
You could almost imagine it's splitting the second and

229
00:10:06,549 --> 00:10:08,240
fourth quadrants.

230
00:10:08,240 --> 00:10:09,980
Now I'll do one more.

231
00:10:09,980 --> 00:10:12,159
Let's do this last one right here.

232
00:10:12,159 --> 00:10:16,829
y is equal to 3.75.

233
00:10:16,830 --> 00:10:19,610
Now you're saying, gee, we're looking for y is

234
00:10:19,610 --> 00:10:21,629
equal to mx plus b.

235
00:10:21,629 --> 00:10:22,939
Where is this x term?

236
00:10:22,940 --> 00:10:24,650
It's completely gone.

237
00:10:24,649 --> 00:10:27,980
Well the reality here is, this could be rewritten as y is

238
00:10:27,980 --> 00:10:33,240
equal to 0x plus 3.75.

239
00:10:33,240 --> 00:10:34,120
Now it makes sense.

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00:10:34,120 --> 00:10:35,500
The slope is 0.

241
00:10:35,500 --> 00:10:39,019
No matter how much we change our x, y does not change.

242
00:10:39,019 --> 00:10:42,929
Delta y over delta x is equal to 0.

243
00:10:42,929 --> 00:10:45,769
I don't care how much you change your x.

244
00:10:45,769 --> 00:10:48,220
Our y-intercept is 3.75.

245
00:10:48,220 --> 00:10:51,870
So 1, 2, 3.75 is right around there.

246
00:10:51,870 --> 00:10:52,710
You want to get close.

247
00:10:52,710 --> 00:10:54,990
3 3/4.

248
00:10:54,990 --> 00:10:57,620
As I change x, y will not change. y is

249
00:10:57,620 --> 00:11:00,570
always going to be 3.75.

250
00:11:00,570 --> 00:11:03,200
It's just going to be a horizontal line at

251
00:11:03,200 --> 00:11:07,450
y is equal to 3.75.

252
00:11:07,450 --> 00:11:10,600
Anyway, hopefully you found this useful.

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