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- So you may or may not already know that any linear equation
- can be written in the form y is equal to mx plus b.
- Where m is the slope of the line.
- The same slope that we've been dealing with
- the last few videos.
- The rise over run of the line.
- Or the inclination of the line.
- And b is the y-intercept.
- I think it's pretty easy to verify that b is a
- y-intercept.
- The way you verify that is you substitute x is equal to 0.
- If you get x is equal to 0-- remember x is equal to 0, that
- means that's where we're going to intercept at the y-axis.
- If x is equal to 0, this equation becomes y is equal to
- m times 0 plus b.
- m times 0 is just going to be 0.
- I don't care what m is.
- So then y is going to be equal to b.
- So the point 0, b is going to be on that line.
- The line will intercept the y-axis at the point
- y is equal to b.
- We'll see that with actual numbers in
- the next few videos.
- Just to verify for you that m is really the slope, let's
- just try some numbers out.
- We know the point 0, b is on the line.
- What happens when x is equal to 1?
- You get y is equal to m times 1.
- Or it's equal to m plus b.
- So we also know that the point 1, m plus b
- is also on the line.
- Right?
- This is just the y value.
- So what's the slope between that point and that point?
- Let's take this as the end point, so you have m plus b,
- our change in y, m plus b minus b over our change in x,
- over 1 minus 0.
- This is our change in y over change in x.
- We're using two points.
- That's our end point.
- That's our starting point.
- So if you simplify this, b minus b is 0.
- 1 minus 0 is 1.
- So you get m/1, or you get it's equal to m.
- So hopefully you're satisfied and hopefully I didn't confuse
- you by stating it in the abstract with all of these
- variables here.
- But this is definitely going to be the slope and this is
- definitely going to be the y-intercept.
- Now given that, what I want to do in this exercise is look at
- these graphs and then use the already drawn graphs to figure
- out the equation.
- So we're going to look at these, figure out the slopes,
- figure out the y-intercepts and then know the equation.
- So let's do this line A first. So what is A's slope?
- Let's start at some arbitrary point.
- Let's start right over there.
- We want to get even numbers.
- If we run one, two, three.
- So if delta x is equal to 3.
- Right?
- One, two, three.
- Our delta y-- and I'm just doing it because I want to hit
- an even number here-- our delta y is equal to-- we go
- down by 2-- it's equal to negative 2.
- So for A, change in y for change in x.
- When our change in x is 3, our change in y is negative 2.
- So our slope is negative 2/3.
- When we go over by 3, we're going to go down by 2.
- Or if we go over by 1, we're going to go down by 2/3.
- You can't exactly see it there, but you definitely see
- it when you go over by 3.
- So that's our slope.
- We've essentially done half of that problem.
- Now we have to figure out the y-intercept.
- So that right there is our m.
- Now what is our b?
- Our y-intercept.
- Well where does this intersect the y-axis?
- Well we already said the slope is 2/3.
- So this is the point y is equal to 2.
- When we go over by 1 to the right, we would have
- gone down by 2/3.
- So this right here must be the point 1 1/3.
- Or another way to say it, we could say it's 4/3.
- That's the point y is equal to 4/3.
- Right there.
- A little bit more than 1.
- About 1 1/3.
- So we could say b is equal to 4/3.
- So we'll know that the equation is y is equal to m,
- negative 2/3, x plus b, plus 4/3.
- That's equation A.
- Let's do equation B.
- Hopefully we won't have to deal with as
- many fractions here.
- Equation B.
- Let's figure out its slope first. Let's start at some
- reasonable point.
- We could start at that point.
- Let me do it right here.
- B.
- Equation B.
- When our delta x is equal to-- let me write it
- this way, delta x.
- So our delta x could be 1.
- When we move over 1 to the right, what happens
- to our delta y?
- We go up by 3.
- delta x.
- delta y.
- Our change in y is 3.
- So delta y over delta x, When we go to the right, our
- change in x is 1.
- Our change in y is positive 3.
- So our slope is equal to 3.
- What is our y-intercept?
- Well, when x is equal to 0, y is equal to 1.
- So b is equal to 1.
- So this was a lot easier.
- Here the equation is y is equal to 3x plus 1.
- Let's do that last line there.
- Line C Let's do the y-intercept first. You see
- immediately the y-intercept-- when x is equal to 0, y is
- negative 2.
- So b is equal to negative 2.
- And then what is the slope?
- m is equal to change in y over change in x.
- Let's start at that y-intercept.
- If we go over to the right by one, two, three, four.
- So our change in x is equal to 4.
- What is our change in y?
- Our change in y is positive 2.
- So change in y is 2 when change in x is 4.
- So the slope is equal to 1/2, 2/4.
- So the equation here is y is equal to 1/2 x, that's our
- slope, minus 2.
- And we're done.
- Now let's go the other way.
- Let's look at some equations of lines knowing that this is
- the slope and this is the y-intercept-- that's the m,
- that's the b-- and actually graph them.
- Let's do this first line.
- I already started circling it in orange.
- The y-intercept is 5.
- When x is equal to 0, y is equal to 5.
- You can verify that on the equation.
- So when x is equal to 0, y is equal to one, two, three,
- four, five.
- That's the y-intercept and the slope is 2.
- That means when I move 1 in the x-direction, I move up 2
- in the y-direction.
- If I move 1 in the x-direction, I move up 2 in
- the y-direction.
- If I move 1 in the x-direction, I move up 2 in
- the y-direction.
- If I move back 1 in the x-direction, I move down 2 in
- the y-direction.
- If I move back 1 in the x-direction, I move down 2 in
- the y-direction.
- I keep doing that.
- So this line is going to look-- I can't draw lines too
- neatly, but this is going to be my best shot.
- It's going to look something like that.
- It'll just keep going on, on and on and on.
- So that's our first line.
- I can just keep going down like that.
- Let's do this second line.
- y is equal to negative 0.2x plus 7.
- Let me write that. y is equal to negative 0.2x plus 7.
- It's always easier to think in fractions.
- So 0.2 is the same thing as 1/5.
- We could write y is equal to negative 1/5 x plus 7.
- We know it's y-intercept at 7.
- So it's one, two, three, four, five, six.
- That's our y-intercept when x is equal to 0.
- This tells us that for every 5 we move to the right,
- we move down 1.
- We can view this as negative 1/5.
- The delta y over delta x is equal to negative 1/5.
- For every 5 we move to the right, we move down 1.
- So every 5.
- One, two, three, four, five.
- We moved 5 to the right.
- That means we must move down 1.
- We move 5 to the right.
- One, two, three, four, five.
- We must move down 1.
- If you go backwards, if you move 5 backwards-- instead of
- this, if you view this as 1 over negative 5.
- These are obviously equivalent numbers.
- If you go back 5-- that's negative 5.
- One, two, three, four, five.
- Then you move up 1.
- If you go back 5-- one, two, three, four,
- five-- you move up 1.
- So the line is going to look like this.
- I have to just connect the dots.
- I think you get the idea.
- I just have to connect those dots.
- I could've drawn it a little bit straighter.
- Now let's do this one, y is equal to negative x.
- Where's the b term?
- I don't see any b term.
- You remember we're saying y is equal to mx plus b.
- Where is the b?
- Well, the b is 0.
- You could view this as plus 0.
- Here is b is 0.
- When x is 0, y is 0.
- That's our y-intercept, right there at the origin.
- And then the slope-- once again you see a negative sign.
- You could view that as negative 1x plus 0.
- So slope is negative 1.
- When you move to the right by 1, when change in x is 1,
- change in y is negative 1.
- When you move up by 1 in x, you go down by 1 in y.
- Or if you go down by 1 in x, you're going to go up by 1 in
- y. x and y are going to have opposite signs.
- They go in opposite directions.
- So the line is going to look like that.
- You could almost imagine it's splitting the second and
- fourth quadrants.
- Now I'll do one more.
- Let's do this last one right here.
- y is equal to 3.75.
- Now you're saying, gee, we're looking for y is
- equal to mx plus b.
- Where is this x term?
- It's completely gone.
- Well the reality here is, this could be rewritten as y is
- equal to 0x plus 3.75.
- Now it makes sense.
- The slope is 0.
- No matter how much we change our x, y does not change.
- Delta y over delta x is equal to 0.
- I don't care how much you change your x.
- Our y-intercept is 3.75.
- So 1, 2, 3.75 is right around there.
- You want to get close.
- 3 3/4.
- As I change x, y will not change. y is
- always going to be 3.75.
- It's just going to be a horizontal line at
- y is equal to 3.75.
- Anyway, hopefully you found this useful.