Video Transcript: Writing Slope-Intercept Equation

Slope-intercept equation from graph 

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. 

Slope-intercept equation from slope & point 

A line has a slope of negative 3/4 and goes through the point 0 comma 8. What is the equation  of this line in slope-intercept form? So any line can be represented in slope-intercept form, is y  is equal to mx plus b, where this m right over here, that is of the slope of the line. And this b  over here, this is the y-intercept of the line. Let me draw a quick line here just so that we can  visualize that a little bit. So that is my y-axis. And then that is my x-axis. And let me draw a line.  And since our line here has a negative slope, I'll draw a downward sloping line. So let's say our  line looks something like that. So hopefully, we're a little familiar with the slope already. The  slope essentially tells us, look, start at some point on the line, and go to some other point of the line, measure how much you had to move in the x direction, that is your run, and then measure  how much you had to move in the y direction, that is your rise. And our slope is equal to rise  over run. And you can see over here, we'd be downward sloping. Because if you move in the  positive x direction, we have to go down. If our run is positive, our rise here is negative. So this  would be a negative over a positive, it would give you a negative number. That makes sense,  because we're downward sloping. The more we go down in this situation, for every step we  move to the right, the more downward sloping will be, the more of a negative slope we'll have.  So that's slope right over here. The y-intercept just tells us where we intercept the y-axis. So  the y-intercept, this point right over here, this is where the line intersects with the y-axis. This  will be the point 0 comma b. And this actually just falls straight out of this equation. When x is  equal to 0-- so let's evaluate this equation, when x is equal to 0. y will be equal to m times 0 plus b. Well, anything times 0 is 0. So y is equal to 0 plus b, or y will be equal to b, when x is equal to  0. So this is the point 0 comma b. Now, they tell us what the slope of this line is. They tell us a  line has a slope of negative 3/4. So we know that our slope is negative 3/4, and they tell us that  the line goes through the point 0 comma 8. They tell us we go through the-- Let me just, in a  new color. I've already used orange, let me use this green color. They tell us what we go  through the point 0 comma 8. Notice, x is 0. So we're on the y-axis. When x is 0, we're on the y axis. So this is our y-intercept. So b, we could say-- we could do a couple-- our y-intercept is the  point 0 comma 8, or we could say that b-- Remember, it's also 0 comma b. We could say b is  equal to 8. So we know m is equal to negative 3/4, b is equal to 8, so we can write the equation  of this line in slope-intercept form. It's y is equal to negative 3/4 times x plus b, plus 8. And we  are done. 

Slope-intercept equation from two points 

A line goes through the points (-1, 6) and (5, -4). What is the equation of the line? Let's just try  to visualize this. So that is my x axis. And you don't have to draw it to do this problem but it  always help to visualize That is my y axis. And the first point is (-1,6) So (-1, 6). So negative 1  coma, 1, 2, 3, 4 ,5 6. So it's this point, rigth over there, it's (-1, 6). And the other point is (5, -4).  So 1, 2, 3, 4, 5. And we go down 4, So 1, 2, 3, 4 So it's right over there. So the line connects them will looks something like this. Line will draw a rough approximation. I can draw a straighter  than that. I will draw a dotted line maybe Easier do dotted line. So the line will looks something 

like that. So let's find its equation. So good place to start is we can find its slope. Remember, we want, we can find the equation y is equal to mx plus b. This is the slope-intercept form where m  is the slope and b is the y-intercept. We can first try to solve for m. We can find the slope of this  line. So m, or the slope is the change in y over the change in x. Or, we can view it as the y value  of our end point minus the y value of our starting point over the x-value of our end point minus  

the x-value of our starting point. Let me make that clear. So this is equal to change in y over  change in x wich is the same thing as rise over run wich is the same thing as the y-value of your  ending point minus the y-value of your starting point. This is the same exact thing as change in  y and that over the x value of your ending point minus the x-value of your starting point This is  the exact same thing as change in x. And you just have to pick one of these as the starting point and one as the ending point. So let's just make this over here our starting point and make that  our ending point. So what is our change in y? So our change in y, to go we started at y is equal  to six, we started at y is equal to 6. And we go down all the way to y is equal to negative 4 So  this is rigth here, that is our change in y You can look at the graph and say, oh, if I start at 6 and I go to negative 4 I went down 10. or if you just want to use this formula here it will give you the  same thing We finished at negative 4, we finished at negative 4 and from that we want to  subtract, we want to subtract 6. This right here is y2, our ending y and this is our beginning y  This is y1. So y2, negative 4 minus y1, 6. or negative 4 minus 6. That is equal to negative 10. And all it does is tell us the change in y you go from this point to that point We have to go down, our  rise is negative we have to go down 10. That's where the negative 10 comes from. Now we just  have to find our change in x. So we can look at this graph over here. We started at x is equal to  negative 1 and we go all the way to x is equal to 5. So we started at x is equal to negative 1, and  we go all the way to x is equal to 5. So it takes us one to go to zero and then five more. So are  change in x is 6. You can look at that visually there or you can use this formula same exact idea,  our ending x-value, our ending x-value is 5 and our starting x-value is negative 1. 5 minus  negative 1. 5 minus negative 1 is the same thing as 5 plus 1. So it is 6. So our slope here is  negative 10 over 6. wich is the exact same thing as negative 5 thirds. as negative 5 over 3 I  divided the numerator and the denominator by 2. So we now know our equation will be y is  equal to negative 5 thirds, that's our slope, x plus b. So we still need to solve for y-intercept to  get our equation. And to do that, we can use the information that we know in fact we have  several points of information We can use the fact that the line goes through the point (-1,6) you  could use the other point as well. We know that when is equal to negative 1, So y is eqaul to 6.  So y is equal to six when x is equal to negative 1 So negative 5 thirds times x, when x is equal to  negative 1 y is equal to 6. So we literally just substitute this x and y value back into this and  know we can solve for b. So let's see, this negative 1 times negative 5 thirds. So we have 6 is  equal to positive five thirds plus b. And now we can subtract 5 thirds from both sides of this  equation. so we have subtracted the left hand side. From the left handside and subtracted from the rigth handside And then we get, what's 6 minus 5 thirds. So that's going to be, let me do it  over here We take a common denominator. So 6 is the same thing as Let's do it over here. So 6  minus 5 over 3 is the same thing as 6 is the same thing as 18 over 3 minus 5 over 3 6 is 18 over 3.  And this is just 13 over 3. And this is just 13 over 3. And then of course, these cancel out. So we  get b is equal to 13 thirds. So we are done. We know We know the slope and we know the y intercept. The equation of our line is y is equal to negative 5 thirds x plus our y-intercept which 

is 13 which is 13 over 3. And we can write these as mixed numbers. if it's easier to visualize. 13  over 3 is four and 1 thirds. So this y-intercept right over here. this y-intercept right over here.  That's 0 coma 13 over 3 or 0 coma 4 and 1 thirds. And even with my very roughly drawn diagram it those looks like this. And the slope negative 5 thirds that's the same thing as negative 1 and 2  thirds. You can see here the slope is downward because the slope is negative. It's a little bit  steeper than a slope of 1. It's not quite a negative 2. It's negative 1 and 2 thirds. if you write this  as a negative, as a mixed number. So, hopefully, you found that entertaining. 

Constructing linear equations from context 

Tara was hiking up a mountain. She started her hike at an elevation of 1,200 meters and  ascended at a constant rate. After four hours, she reached an elevation of 1,700 meters. Let y  represent Tara's elevation in meters after x hours. And they ask us, and this is from an exercise  on Khan Academy, it says, complete the equation for the relationship between the elevation  and the number of hours. And if you're on Khan Academy, you would type it in, but we can do it by hand. So pause this video and work it out on some paper and let's see if we get to the same  place. All right, now let's do this together. So first of all, they tell us that she's ascending at a  constant rate. So that's a pretty good indication that we could describe her elevation based on  the number of hours she travels with a linear equation. And we could even figure out that  constant rate. It says that she goes from 1,200 meters to 1,700 meters in four hours. So we  could say her rate is going to be her change in elevation over a change in time. So her change in elevation is 1,700 meters minus 1,200 meters and she does this over four hours. Over, her  change in time is four hours. So her constant rate in the numerator here, 1,700 minus 1,200 is  500 meters. She's able to go up 500 meters in four hours. If we divide 500 by four, this is 125  meters per hour. And so we could use this now to think about what our equation would be. Our  elevation y would be equal to, well, where is she starting? Well, it's starting at 1,200 meters. So  she's starting at 1,200 meters. And then to that, we're going to add how much she climbs based on how many hours she's traveled. So it's going to be this rate, 125 meters per hour times the  number of hours she has been hiking. So the number of hours is x times x. So this right over  here is an equation for the relationship between the elevation and the number of hours.  Another way you could have thought about it, you could have said, okay, this is going to be a  linear equation because she's ascending at a constant rate. You could say the slope intercept  form for a linear equation is y is equal to mx plus b, where b is your y-intercept. What is the  value of y when x is equal to zero? And you'd say, all right, when x is equal to zero, she's at an  elevation of 1,200. And then m is our slope. So that's the rate at which our elevation is  increasing. And that's what we calculated right over here. Our slope is 125 meters per hour. So  notice, these are equivalent. I just have, these two terms are swapped. So we could either write  y is equal to 1,200 plus 125x or you could write it the other way around. You could write 125x  plus 1,200. They are equivalent.