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- In this video I'm going to do a bunch of examples of finding
- the equations of lines in slope-intercept form.
- Just as a bit of a review, that means equations of lines
- in the form of y is equal to mx plus b where m is the slope
- and b is the y-intercept.
- So let's just do a bunch of these problems. So here they
- tell us that a line has a slope of negative 5, so m is
- equal to negative 5.
- And it has a y-intercept of 6.
- So b is equal to 6.
- So this is pretty straightforward.
- The equation of this line is y is equal to
- negative 5x plus 6.
- That wasn't too bad.
- Let's do this next one over here.
- The line has a slope of negative 1 and contains the
- point 4/5 comma 0.
- So they're telling us the slope, slope of negative 1.
- So we know that m is equal to negative 1, but we're not 100%
- sure about where the y-intercept is just yet.
- So we know that this equation is going to be of the form y
- is equal to the slope negative 1x plus b, where b is the
- y-intercept.
- Now, we can use this coordinate information, the
- fact that it contains this point, we can use that
- information to solve for b.
- The fact that the line contains this point means that
- the value x is equal to 4/5, y is equal to 0 must satisfy
- this equation.
- So let's substitute those in. y is equal to 0 when x is
- equal to 4/5.
- So 0 is equal to negative 1 times 4/5 plus b.
- I'll scroll down a little bit.
- So let's see, we get a 0 is equal to negative 4/5 plus b.
- We can add 4/5 to both sides of this equation.
- So we get add a 4/5 there.
- We could add a 4/5 to that side as well.
- The whole reason I did that is so that cancels out with that.
- You get b is equal to 4/5.
- So we now have the equation of the line.
- y is equal to negative 1 times x, which we write as negative
- x, plus b, which is 4/5, just like that.
- Now we have this one.
- The line contains the point 2 comma 6 and 5 comma 0.
- So they haven't given us the slope or the y-intercept
- explicitly.
- But we could figure out both of them from these
- coordinates.
- So the first thing we can do is figure out the slope.
- So we know that the slope m is equal to change in y over
- change in x, which is equal to-- What is the change in y?
- Let's start with this one right here.
- So we do 6 minus 0.
- Let me do it this way.
- So that's a 6-- I want to make it color-coded-- minus 0.
- So 6 minus 0, that's our change in y.
- Our change in x is 2 minus 2 minus 5.
- The reason why I color-coded it is I wanted to show you
- when I used this y term first, I used the 6 up here, that I
- have to use this x term first as well.
- So I wanted to show you, this is the coordinate 2 comma 6.
- This is the coordinate 5 comma 0.
- I couldn't have swapped the 2 and the 5 then.
- Then I would have gotten the negative of the answer.
- But what do we get here?
- This is equal to 6 minus 0 is 6.
- 2 minus 5 is negative 3.
- So this becomes negative 6 over 3, which is the same
- thing as negative 2.
- So that's our slope.
- So, so far we know that the line must be, y is equal to
- the slope-- I'll do that in orange-- negative 2 times x
- plus our y-intercept.
- Now we can do exactly what we did in the last problem.
- We can use one of these points to solve for b.
- We can use either one.
- Both of these are on the line, so both of these must satisfy
- this equation.
- I'll use the 5 comma 0 because it's always nice when
- you have a 0 there.
- The math is a little bit easier.
- So let's put the 5 comma 0 there.
- So y is equal to 0 when x is equal to 5.
- So y is equal to 0 when you have negative 2 times 5, when
- x is equal to 5 plus b.
- So you get 0 is equal to -10 plus b.
- If you add 10 to both sides of this equation, let's add 10 to
- both sides, these two cancel out.
- You get b is equal to 10 plus 0 or 10.
- So you get b is equal to 10.
- Now we know the equation for the line.
- The equation is y-- let me do it in a new color-- y is equal
- to negative 2x plus b plus 10.
- We are done.
- Let's do another one of these.
- All right, the line contains the points 3 comma 5 and
- negative 3 comma 0.
- Just like the last problem, we start by figuring out the
- slope, which we will call m.
- It's the same thing as the rise over the run, which is
- the same thing as the change in y over the change in x.
- If you were doing this for your homework, you wouldn't
- have to write all this.
- I just want to make sure that you understand that these are
- all the same things.
- Then what is our change in y over our change in x?
- This is equal to, let's start with the side first. It's just
- to show you I could pick either of these points.
- So let's say it's 0 minus 5 just like that.
- So I'm using this coordinate first. I'm kind of viewing it
- as the endpoint.
- Remember when I first learned this, I would always be
- tempted to do the x in the numerator.
- No, you use the y's in the numerator.
- So that's the second of the coordinates.
- That is going to be over negative 3 minus 3.
- This is the coordinate negative 3, 0.
- This is the coordinate 3, 5.
- We're subtracting that.
- So what are we going to get?
- This is going to be equal to-- I'll do it in a neutral
- color-- this is going to be equal to the numerator is
- negative 5 over negative 3 minus 3 is negative 6.
- So the negatives cancel out.
- You get 5/6.
- So we know that the equation is going to be of the form y
- is equal to 5/6 x plus b.
- Now we can substitute one of these coordinates in for b.
- So let's do.
- I always like to use the one that has the 0 in it.
- So y is a zero when x is negative 3 plus b.
- So all I did is I substituted negative 3 for x, 0 for y.
- I know I can do that because this is on the line.
- This must satisfy the equation of the line.
- Let's solve for b.
- So we get zero is equal to, well if we divide negative 3
- by 3, that becomes a 1.
- If you divide 6 by 3, that becomes a 2.
- So it becomes negative 5/2 plus b.
- We could add 5/2 to both sides of the equation,
- plus 5/2, plus 5/2.
- I like to change my notation just so you get
- familiar with both.
- So the equation becomes 5/2 is equal to-- that's a 0-- is
- equal to b.
- b is 5/2.
- So the equation of our line is y is equal to 5/6 x plus b,
- which we just figured out is 5/2, plus 5/2.
- We are done.
- Let's do another one.
- We have a graph here.
- Let's figure out the equation of this graph.
- This is actually, on some level, a little bit easier.
- What's the slope?
- Slope is change in y over change it x.
- So let's see what happens.
- When we move in x, when our change in x is 1, so that is
- our change in x.
- So change in x is 1.
- I'm just deciding to change my x by 1, increment by 1.
- What is the change in y?
- It looks like y changes exactly by 4.
- It looks like my delta y, my change in y, is equal to 4
- when my delta x is equal to 1.
- So change in y over change in x, change in y is 4 when
- change in x is 1.
- So the slope is equal to 4.
- Now what's its y-intercept?
- Well here we can just look at the graph.
- It looks like it intersects y-axis at y is equal to
- negative 6, or at the point 0, negative 6.
- So we know that b is equal to negative 6.
- So we know the equation of the line.
- The equation of the line is y is equal to the slope times x
- plus the y-intercept.
- I should write that.
- So minus 6, that is plus negative 6 So that is the
- equation of our line.
- Let's do one more of these.
- So they tell us that f of 1.5 is negative 3, f of
- negative 1 is 2.
- What is that?
- Well, all this is just a fancy way of telling you that the
- point when x is 1.5, when you put 1.5 into the function, the
- function evaluates as negative 3.
- So this tells us that the coordinate 1.5, negative 3 is
- on the line.
- Then this tells us that the point when x is negative 1, f
- of x is equal to 2.
- This is just a fancy way of saying that both of these two
- points are on the line, nothing unusual.
- I think the point of this problem is to get you familiar
- with function notation, for you to not get intimidated if
- you see something like this.
- If you evaluate the function at 1.5, you get negative 3.
- So that's the coordinate if you imagine that y is
- equal to f of x.
- So this would be the y-coordinate.
- It would be equal to negative 3 when x is 1.5.
- Anyway, I've said it multiple times.
- Let's figure out the slope of this line.
- The slope which is change in y over change in x is equal to,
- let's start with 2 minus this guy, negative 3-- these are
- the y-values-- over, all of that over, negative
- 1 minus this guy.
- Let me write it this way, negative 1 minus
- that guy, minus 1.5.
- I do the colors because I want to show you that the negative
- 1 and the 2 are both coming from this, that's why I use
- both of them first. If I used these guys first, I would have
- to use both the x and the y first. If I use the 2 first, I
- have to use the negative 1 first. That's why I'm
- color-coding it.
- So this is going to be equal to 2 minus negative 3.
- That's the same thing as 2 plus 3.
- So that is 5.
- Negative 1 minus 1.5 is negative 2.5.
- 5 divided by 2.5 is equal to 2.
- So the slope of this line is negative 2.
- Actually I'll take a little aside to show you it doesn't
- matter what order I do this in.
- If I use this coordinate first, then I have to use that
- coordinate first. Let's do it the other way.
- If I did it as negative 3 minus 2 over 1.5 minus
- negative 1, this should be minus the 2 over 1.5 minus the
- negative 1.
- This should give me the same answer.
- This is equal to what?
- Negative 3 minus 2 is negative 5 over 1.5 minus negative 1.
- That's 1.5 plus 1.
- That's over 2.5.
- So once again, this is equal the negative 2.
- So I just wanted to show you, it doesn't matter which one
- you pick as the starting or the endpoint, as long as
- you're consistent.
- If this is the starting y, this is the starting x.
- If this is the finishing y, this has to be
- the finishing x.
- But anyway, we know that the slope is negative 2.
- So we know the equation is y is equal to negative 2x plus
- some y-intercept.
- Let's use one of these coordinates.
- I'll use this one since it doesn't have a decimal in it.
- So we know that y is equal to 2.
- So y is equal to 2 when x is equal to negative 1.
- Of course you have your plus b.
- So 2 is equal to negative 2 times negative 1 is 2 plus b.
- If you subtract 2 from both sides of this equation, minus
- 2, minus 2, you're subtracting it from both sides of this
- equation, you're going to get 0 on the left-hand side is
- equal to b.
- So b is 0.
- So the equation of our line is just y is
- equal to negative 2x.
- Actually if you wanted to write it in function notation,
- it would be that f of x is equal to negative 2x.
- I kind of just assumed that y is equal to f of x.
- But this is really the equation.
- They never mentioned y's here.
- So you could just write f of x is equal to 2x right here.
- Each of these coordinates are the coordinates
- of x and f of x.
- So you could even view the definition of slope as change
- in f of x over change in x.
- These are all equivalent ways of viewing the same thing.