Video Transcript: Intro to Slope-Intercept Form

Intro to slope-intercept form 

There's a lot of different ways that you could represent a linear equation. So for example, if you  had the linear equation y is equal to 2x plus three, that's one way to represent it, but I could  represent this in an infinite number of ways. I could, let's see, I could subtract 2x from both  sides, I could write this as negative 2x plus y is equal to three. I could manipulate it in ways  where I get it to, and I'm gonna do it right now, but this is another way of writing that same  thing. y minus five is equal to two times x minus one. You could actually simplify this and you  could get either this equation here or that equation up on top. These are all equivalent, you can  get from one to the other with logical algebraic operations. So there's an infinite number of  ways to represent a given linear equation, but I what I wanna focus on in this video is this  representation in particular, because this one is a very useful representation of a linear equation and we'll see in future videos, this one and this one can also be useful, depending on what you  are looking for, but we're gonna focus on this one, and this one right over here is often called  slope-intercept form. Slope-intercept form. And hopefully in a few minutes, it will be obvious  why it called slop-intercept form. And before I explain that to you, let's just try to graph this  thing. I'm gonna try to graph it, I'm just gonna plot some points here, so x comma y, and I'm  gonna pick some x values where it's easy to calculate the y values. So maybe the easiest is if x is equal to zero. If x is equal to zero, then two times zero is zero, that term goes away, and you're  only left with this term right over here, y is equal to three. Y is equal to three. And so if we were  to plot this. Actually let me start plotting it, so that is my y axis, and let me do the x axis, so that can be my x, oh that's not as straight as I would like it. So that looks pretty good, alright. That is my x axis and let me mark off some hash marks here, so this is x equals one, x equals two, x  equals three, this is y equals one, y equals two, y equals three, and obviously I could keep going  and keep going, this would be y is equal to negative one, this would be x is equal to negative  one, negative two, negative three, so on and so forth. So this point right over here, zero  comma three, this is x is zero, y is three. Well, the point that represents when x is equal to zero  and y equals three, this is, we're right on the y axis. If they have a line going through it and this  line contains this point, this is going to be the y- intercept. So one way to think about it, the  reason why this is called slope-intercept form is it's very easy to calculate the y-intercept. The  y-intercept here is going to happen when it's written in this form, it's going to happen when x is  equal to zero and y is equal to three, it's gonna be this point right over here. So it's very easy to  figure out the intercept, the y-intercept from this form. Now you might be saying, well it says  slope-intercept form, it must also be easy to figure out the slope from this form. And if you  made that conclusion, you would be correct! And we're about to see that in a few seconds. So  let's plot some more points here and I'm just gonna keep increasing x by one. So if you increase  x by one, so we could write that our delta x, our change in x, delta Greek letter, this triangle is a  Greek letter, delta, represents change in. Change in x here is one. We just increased x by one,  what's gonna be our corresponding change in y? What's going to be our change in y? So let's  see, when x is equal to one, we have two times one, plus three is going to be five. So our  change in y is going to be two. Let's do that again. Let's increase our x by one. Change in x is  equal to one. So then if we're gonna increase by one, we're gonna go from x equals one to x  equals two. Well what's our corresponding change in y? Well when x is equal to two, two times  two is four, plus three is seven. Well our change in y, our change in y is equal to two. Went from 

five- when x went from one to two, y went from five to seven. So for every one that we increase x, y is increasing by two. So for this linear equation, our change in y over change in x is always  going to be, our change in y is two when our change in x is one, or it's equal to two, or we could  say that our slope is equal to two. Well let's just graph this to make sure that we understand  this. So when x equals one, y is equal to five. And actually we're gonna have to graph five up  here. So when x is equal to one, y is equal to, and actually this is a little bit higher, this, let me  clean this up a little bit. So this one would be, erase that a little bit. Just like that. So that's y is  equal to four, and this is y is equal to five. So when x is one, y is equal to five, so it's that point  right over there. So our line is going to look- you only need two points to define a line, our line  is going to like, let me do this in this color right over here. Our line is going to look like, is going  to look, is going to look something like, is going to look, let me see if I can, I didn't draw it  completely at scale, but it's going to look something like this. This is the line, this is the line, y is  equal to 2x plus three. But we already figured out that its slope is equal to two, when our  change in x is one, when our change in x is one, our change in y is two. If our change in x was  negative one, if our change in x was negative one, our change in y is negative two. And you can  see that, if from zero we went, we went down one, if we went to negative one, then what's our  y going to be? Two times negative one is negative two plus three is one. So we see that, the  point negative one comma one is on the line as well. So the slope here, our change in y over  change in x, if we're going from between any two points on this line, is always going to be two.  But where do you see two in this original equation? Well you see the two right over here. And  when you write something in slop-intercept form, where you explicitly solve for y, y is equal to  some constant times x to the first power plus some other constant, the second one is going to  be your intercept, your y-intercept, or it's going to be a way to figure out the y-intercept, the  intercept itself is this point, the point at which the line intercepts the y axis, and then this two is going to represent your slope. And that makes sense because every time you increase x by one, you're gonna multiply that by two, so you're gonna increase y by two. So this is just a, kinda of  a get your feet wet with the idea of slope-intercept form, but you'll see, at least for me, this is  the easiest form for me to think about what the graph of something looks like, because if you  were given another, if you were given another linear equation, let's say y is equal to negative x,  negative x plus two. Well immediately you say, okay look, my yintercept is going to be the point zero comma two, so I'm gonna intersect the y axis right at that point, and then I have a slope of, the coefficient here is really just negative one, so I have a slope of negative one. So as we  increase x by one, we're gonna decrease y by one. Increase x by one, you're gonna decrease y  by one. If you increase x by two, you're gonna decrease y by two. And so our line is gonna look  something like this. Let me see if I can draw it relatively neatly. It's going to look something, I  think I can do a little bit better than that. It's 'cus my graph paper is hand drawn. It's not ideal,  but I think you get, you get the point. It's gonna look something like that. So from slope intercept form, very easy to figure out what the y-intercept is, and very easy to figure out the  slope. The slope here, slope here is negative one. That's this negative one right over here, and  the y-intercept, y-intercept is the point zero comma two, very easy to figure out 'cus essentially that gave you the information right there. 

Slope and y-intercept from equation

What I'd like to do in this video is a few more examples recognizing the slope and y-intercept  given an equation. So, let's start with something that we might already recognize. Let's say we  have something of the form y is equal to five x plus three. What is the slope and the y-intercept  in this example here? Well, we've already talked about that we can have something in slope  intercept form where it has a form y is equal to the slope, which people use the letter m for, the  slope times x plus the y-intercept, which people use the letter b for. So, if we just look at this, m is going to be the coefficient on x right over there. So, m is equal to five. That is the slope. And b is just going to be this constant term, plus three. So, b is equal to three. So, this is your y intercept. So, that's pretty straightforward but let's see a few slightly more involved examples.  Let's say if we had form y is equal to five plus three x, what is the slope and the y-intercept in  this situation? 

Worked examples: slope-intercept intro 

Let's do some practice examples from our intro to slope y intercept exercise. What is the slope  of y is equal to negative four x minus three? So you might already recognize, this is in slope  intercept form. Just as a reminder, slope intercept form is y is equal to mx plus b, where the  coefficient on this x term right over here, that is our slope, and then this constant right over  here, that is going to give you your y intercept. So if they're saying, what is the slope here? Well  I just need to figure out, what is the coefficient on this x term? And you can see that the  coefficient here is a negative four. So that is going to be our m, that is going to be our slope.  Now just as a reminder, you have to make sure that it's solved in this way, that is is solved for y,  y is equal to something times x minus three, so that's our slope. Let's do another one of these.  So we're asked, what is the y intercept of y is equal to negative three x minus two? So once  again we already have it in slope intercept form. It's already been solved for y. It's of the form y  is equal to mx plus b, where m, our slope, is given right over here, negative three. But they're  not asking for our slope, they're asking for the y intercept. Well the y intercept is given by b  here, so b is negative two. Pay close attention to the sign here. So b is equal to negative two.  But when I look at these choices, I don't see a b is equal to negative two, so what are they  talking about? Well a y intercept is, what is the y value when x is equal to zero? And you could  see that here, if x was equal to zero then that term goes away and y is equal to b. So if you want to know the point where the graph described by this equation intercepts the y axis, well it's  going to be what is y when x is equal to zero? Well when x is equal to zero, y is equal to negative two. And you could see that in our original equation, again. If x were zero, this term would go  away, and y would be equal to negative two. So zero, comma, negative two. So it would be that choice right over there. On Khan Academy obviously you just have to click on that, you don't  have to shade it in. Let's do one more. Complete the equation of the line whose slope is five,  and y intercept is zero, comma, four. So once again the general form is y is equal to our slope  times x, if I want to put it in slope intercept form, plus our y intercept. Well they're telling us,  our slope is five. Whose slope is five. So we know that m is going to be five. And they tell us that the y intercept is zero, four. So the y intercept, b, that is the value of y when x is equal to zero.  So the value of y when x equals zero is this four right over here. So that is going to be four. So I  could say y is equal to five times x plus four. And when you're actually entering it on Khan  Academy, you would just type it in, or if you're using the app you would use it with your finger. 

And I always make the mistake of writing y equals, and I type in y equals five x plus four. Notice  they already gave you the y equals right over there. That's all you have to do, recognize the  slope, the y intercept, and then remember what the slope intercept form actually is. 

Linear equation word problems 

When Quinn returned from vacation, he turned the heat back on in his home. He set the  temperature as high as it could go. Q represents the temperature in Quinn's home in degrees  Celsius after t minutes. This Q is equal to 15 plus 0.4t. What was the temperature when Quinn  returned from vacation? So pause this video and see if you can work this out on your own. All  right, so they wanna know the temperature and you might get a little confused, hey maybe t is  for temperature. No, t is time in minutes. Temperature is Q. Q represents the temperature. So  they really wanna know is, what was Q when Quinn returned from vacation? Well, right when  Quinn returned from vacation, that is when t is equal to zero. So this is equivalent to saying,  what is Q, our temperature, when zero minutes have elapsed? Well, if you go back to this  original equation, we see that Q is equal to 15 plus 0.4 times the amount of elapsed time in  minutes. So that's times zero. So that's just going to be 15 degrees Celsius. If you're familiar  with slope intercept form, you could think of it as our temperature is equal to 0.4 times the  elapsed time plus 15. So t equals zero, you're left with just this term which in many cases we  view as our y-intercept. What is going on right when we're just getting started? Right when our  horizontal variable is equal to zero. And our horizontal variable in this situation is elapsed time.  How much do the temperature increase every minute? There's a couple of ways you can think  about this. One if you recognize, this is slope intercept form. You could see that 0.4 is the slope.  So that says for every one minute change in time, you're going to have an increase in  temperature by 0.4 degrees Celsius. So you could do it that way. You could try out some values.  You could say, all right, let me think about what Q is going to be based on t. So time, t equals  zero right when you got home. We're to figure out that the temperature is 15 degrees Celsius.  At t equals one, what happens? It's going to be 15 plus 0.4 times one. That's just gonna be 15.4.  Notice, when we increased our time by one, our temperature increased by 0.4 degrees Celsius  by the slope. And what happened again if we increased time by another minute, if we go from  one to two, we would get two, 15.8. We would increase temperature by another 0.4. How much will the temperature increase if Quinn leaves the heat on for 20 minutes? Pause the video and  see if you can solve that. All right, now we have to be careful here. They're not asking us what is the temperature after 20 minutes. They're saying, how much will the temperature increase if he leaves the heat on for 20 minutes? If we just wanna know what is the temperature after 20  minutes, we would just say, okay, what is Q when t is equal to 20? So it'd be 15 plus 0.4 times  20. 0.4 times 20 is eight. Eight plus 15 is 23. So it's 23 degrees Celsius after 20 minutes. But  that's not what they're asking us. They're asking, how much will the temperature increase?  Well, where did we start from? We started from 15 degrees Celsius and now after 20 minutes  we have gone to 23 degrees Celsius. So we have increased by eight degrees Celsius. Or, another way to think about it, we have increased by this amount right over here. We started at 15 and  after 20 minutes, we have increased by 0.4 times 20 which is eight degrees Celsius. We're done!