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1.  A review of the coordinate plane and plotting points in the first quadrant.

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1.  A review of the coordinate plane and plotting points in the first quadrant.

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- [Instructor] You're probably familiar with the notion of a number line where we can take a number and associate it with a point on the number line. So for example, the number two, I would go, I would start at zero, I'd go one, two to the right, and I would end up right over there. What we're going to do in this video is think about how do we take two numbers and associate them with a point on a plane like this. So for example, you might have the two numbers three, comma five. How can these two numbers be represented as a point on the plane, or how can these tell you where a point on the plane is? So let's first get a little bit of terminology out of the way. So what we have here, this is often known as our coordinate plane. These two numbers you could view as our coordinates. Let me write this down. These are coordinates. These black lines are known as the axes. Each one is an axis and the one that goes left right here, this is known as the X axis, or it's typically known as the X axis. In the future it might be called other things, and the one that goes up down in the vertical direction, this is typically known as the Y axis. As you go further in math we might call it other things, but most of the time it's going to be called the Y axis. So how does three comma five, how do these coordinates specify a point on this plane? Well the way that we typically work it through, the standard way that people will interpret these points, it'll say all right, this first coordinate, this is our X coordinate. This is our X coordinate. You can view it as how far do we move to the right along the X axis. So what you would do is you'd say all right I'm gonna start right here where my axes intersect and I'm gonna go three to the right. One, two, three. So my X coordinate says, all right my point is going to be this far to the right. This far to the right. So it's gonna be somewhere on this vertical line. This dotted line that I'm showing. Everything on this vertical line has an X coordinate of three. Now what's the Y coordinate? Another way I should say it, the second number right over here. This is the Y coordinate. Y coordinate. This tells us how far do we move up. So one way to think about it, you could start back where the axes intersect. This point is actually called the origin. Let me write that down. That is the origin, and so starting at the origin, move five up. One, two, three, four, five. So everything on this horizontal line that I'm drawing has a Y coordinate of five. So what point uniquely has both an X coordinate of three and a Y coordinate of five? Well you can see where those two lines intersect, right over here. Actually let me do that same blue color that I wrote the coordinates in. So this point right over here, it has an X coordinate of three, and it has a Y coordinate of five. That is the point three comma five. Now what is the coordinate of the origin? Well the origin is zero to the right of the origin and it's also zero above the origin. So the coordinates there, the X coordinate is zero, the Y coordinate is also zero. Let's do a few more examples. So let's say that I wanted to plot the point two comma five. Why don't you pause this. Oh let me use a different number. Two comma four. Why don't you pause this video and think about where that point would be on this coordinate plane. All right let's do it together. So the first number is going to be our X coordinate. It tells us how far do we move to the right. So we move two to the right, and then our second number says how far do we move up. So first we're gonna move two to the right and then we are going to move four up. So you can say one two to the right and then one two three four right over there. This right over here is the point two comma four. Notice is X coordinate. How far to the right of the origin it is. That is two and its Y coordinate, how far above the origin is. That is four. Now let's go the other way around. Let's say that I were to give you, if I were to give you this point right over here. What would its coordinates be? Pause the video and try to figure that out. All right, well we know it's gonna be two numbers. So I'll do something comma something. Now the first something, that's going to be our X coordinate. You could think of it, what point on the X axis are we above? You could think about how far to the right of the origin we are, and you can see that your X coordinate right over here is if we just drop a vertical line straight down from that point it hits the X axis at four. So that is four. Another way to think about it we're one two three four to the right of the Y axis. Now how high are we? How high above the X axis are we? Well we're one above the X asis. So this is gonna be four comma one. Another way to think about it, if you just take a line and you go straight to the left you're going to hit the Y axis at the one right over here. So the coordinates here are four comma one. Now just so we don't get confused, and when you first learn this, the main point of confusion is remembering that okay the first number is the X coordinate, the second number is the Y coordinate. One comma four would be a different point. One comma four would be okay one in the X direction and then four in the Y direction. So this is one comma four over here. So it's very important to realize that the standard way of interpreting these numbers is that the first one says how far do you move to the right of the origin, or how far do you move along the X axis or where are you on the X axis, and the second number is how far do you move in the vertical direction, or where are you relative to the Y, or where are you in the vertical direction?

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Introduction to the coordinate plane (video) | Khan Academy

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Instructor] You're probably familiar with the notion of a number line where we can take a number and associate it with a point on the number line. So for example, the number two, I would go, I would start at zero, I'd go one, two to the right, and I would end up right over there.

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2.  Graphing points in the coordinate plane using all four quadrants.

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2.  Graphing points in the coordinate plane using all four quadrants.

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Points on the coordinate plane (practice) | Khan Academy

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In which quadrant is the point negative 7 comma 7 located? So let's just review what a quadrant is. A quadrant are each of the four sections of the coordinate plane. And when we talk about the sections, we're talking about the sections as divided by the coordinate axes. So this right here is the x-axis and this up-down axis is the y-axis. And you can see it divides a coordinate plane into four sections. We call each of these sections quadrants. This one over here, where both the x-values and the y-values are positive, we call the first quadrant. And we use the Roman numeral I. Then if we kind of move counterclockwise around the coordinate plane, this quadrant where the x-values are negative and the y-values are positive, we call this the second quadrant. I could write it. We call this the second quadrant. Then we go down here where both the x-values are negative and the y-values are negative. We call this the third quadrant, once again using Roman numerals. Then finally the quadrant where the x-values are positive but the y-values are negative, we call this the fourth quadrant. So let's see which quadrant the point negative 7 comma 7 is located. So there's two ways to think about it. You could just say, look, we have a negative x-value. Our x-value is negative, so we're going to move to the left. So we're going to be on this side. We're going to be on this side right here of the coordinate plane. Just by the fact that the x-value is negative, we're going to be either in the second or the third quadrant. Now, we know that the y-value is positive. We know that the y-value is positive. So if the x-value is negative and the y-value is positive, we're going to land someplace right over here in the second quadrant. The other way to think about it is you could literally just plot this point and see that it falls in the second quadrant. So let's do that. If x is negative 7. So that's negative 1, negative 2, negative 3, negative 4, negative 5, negative 6, negative 7. Did I do that right? 1, 2, 3 4, 5, 6, 7. So this is x is negative 7. And then we have to go up 7 because y is equal to positive 7. So 1, 2, 3, 4, 5, 6 7. So the point negative 7 comma 7 is right over here, clearly lies in the second quadrant.

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Quadrants of the coordinate plane | Graphs | Math (video) | Khan Academy

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In which quadrant is the point negative 7 comma 7 located? So let's just review what a quadrant is. A quadrant are each of the four sections of the coordinate plane. And when we talk about the sections, we're talking about the sections as divided by the coordinate axes.

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5.  Practice!  Go ahead and get 100% on this one too!

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5.  Practice!  Go ahead and get 100% on this one too!

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Practice graphing points on a coordinate plane and identifying the quadrant or axis where they are located.

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Learn about a class of equations in two variables that's called "linear equations." They are called that way because their graph is a line. These are the most basic and probably most useful equations you will ever know!

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The solution of an equation with one variable is a number. How does the solution of a 2-variable equation look like? Well, it is an ordered pair. Learn more about it and how to test solutions to 2-variable equations.

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Determine which ordered pairs are solutions to equations.

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Complete solutions to 2-variable equations

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Given the x or y value of a 2-variable equation solution, find the value for the other variable in the solution.

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- [Voiceover] Let's say that we have the linear equation, y = 1/2x - 3. So if we wanted to draw the line that represents the set of all points, all the coordinates where the x value and the y value satisfy this equation, we could start off by just trying to draw, by trying to draw a few of those points, and then connecting them with a line. Let's set up a little table here x, y. And we can just try a couple of x values here, then figure out what the corresponding y values are. I'm going to pick x values where it's going to be fairly easy to calculate the y values. Let's say when x is equal to zero, then you're gonna have 1/2 x 0 - 3, well then y is going to be -3. When x is, let me try x = 2, because then 1/2 x 2 is just gonna be 1. So when x = 2, you're going to have 1/2 x 2 = 1, -3 is -2. When x is equal to, let's try 4. So 1/2 x 4 is 2, and then -3 is -1, and we could keep going but actually all we need is two points for a line. So we're ready to plot this line if we'd like. The point 0, -3 is on this line. 0, -3 and actually let me do this in a slightly darker color so that we can see it on this white background. 0, -3 is on the line, 2, -2 is on the line. So 2, -2 and then we have 4, -1. So when x is 4, y is -1, and I could draw a line that connects all of these so it would look something like... If I, let's see if I could do this. It would look something like, it would something like, like that. So this right over here, this is literally, this is the graph of y = 1/2x - 3. Now when we look at a graph like this an interesting thing that we might want to ask ourself is where does the graph intersect our axes? So first we can say, well where does in intersect our x-axis? When you look at this, it looks like it happens at this point right over here. This point where a graph intersects an axes this is called an intercept. This one in particular is called the x-intercept. Why is it called the x-intercept? Because that's where the graph is intersecting the x-axis and the x-intercept, it looks like this is at the .6, 0. Now it's very interesting, the x-intercept happens when y = 0. Remember, you're on the x-axis when you haven't moved up or down from that axis which means y = 0. So your x-intercept happens at x = 6, y = 0. It's this coordinate. Now what about the y-intercept? Well the y-intercept is this point right over here. This is where you intersect or I guess you could say intercept the y-axis. So this right over here, that over there is the y-intercept. The y-intercept is at the coordinate that has a 0 for the x-coordinate. X is 0 here and y is -3. X is 0 and y is -3. This was actually one of the points, or one of the pairs that we first tried out. You can validate that 6, 0 satisfies this equation right over here. If x is 6, 1/2 x 6 is 3, -3 is indeed equal to 0. So now that we know what an x-intercept is, it's the point where a graph intersects the x-axis or intercepts the x-axis and the y-intercept is the point where a graph intercepts the y-axis or intersects the y-axis. Let's try to see if we can find the x and y-intercepts for a few other linear equations. So let's say that I had the linear equation. Let's say that I have 5x + 6y = 30. I encourage you to pause this video, and figure out what are the x and y-intercepts for the graph that represents the solutions, all the xy pairs that satisfy this equation. Well the easiest thing to do here, let's see what the y value is when x = 0 and what x value is when y = 0. When x = 0 this becomes 6y = 30. So 6 times what is 30? Well y would be equal to 5 here. So when x is 0, y is 5. What about when y is 0? Well when y is 0, that's going to be 0, and you have 5x = 30. Well then x would be equal to 6. Then x would be equal to 6. So we could plot those points, 0, 5. When x is 0, y is 5. When x is 6, y is 0. So those are both points on this graph and then the actual graph is going to, or the actual line that represents the x and y pairs that satisfy this equation is going to look like, it's going to look like this. I'll just try. So I can make it go, it's going to look like... It's going to go through those two points. So it going to...I can make it go the other way too. Let me see. It's going to go through those two points and so it's going to look something like that. Now what are its' x and y-intercepts? Well, we already kind of figured it out but the intercepts themselves, these are the points on the graph where they intersect the axes. So this right over here, this is the y-intercept. That point is the y-intercept and it happens, it's always going to happen when x = 0, and when x = 0 we know that y = 5. It's that point, the point 0, 5. And what is the y inter...what is the x-intercept? The x-intercept is the point, it's actually the same x-intercept for this equation right over here. It's the point 6, 0. That point right over there.

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Voiceover] Let's say that we have the linear equation, y = 1/2x - 3. So if we wanted to draw the line that represents the set of all points, all the coordinates where the x value and the y value satisfy this equation, we could start off by just trying to draw, by trying to draw a few of those points, and then connecting them with a line.

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Sal determines the x-intercept of a linear equation from a graph. Afterwards, he checks his work by plugging values back into the equation.

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4.  Intercepts from an equation of a line.

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4.  Intercepts from an equation of a line.

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Sal finds the x and y-intercepts of -5x + 4y = 20.

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5.  Practice!  I know you can do this 100% accurately!

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5.  Practice!  I know you can do this 100% accurately!

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x-intercepts and y-intercepts

x-intercepts and y-intercepts

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Intercepts from an equation (practice) | Khan Academy

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Sal finds the y-intercept of the graph of a linear function given a table of values.

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Find the intercepts of a line given a table of values.

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8.  Intercepts review.  Read through these notes.  Do you got this?

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8.  Intercepts review.  Read through these notes.  Do you got this?

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Intercepts of lines review (x-intercepts and y-intercepts)

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The x-intercept is where a line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis. Thinking about intercepts helps us graph linear equations.

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- [Voiceover] As we start to graph lines, we might notice that they're differences between lines. For example, this pink or this magenta line here, it looks steeper than this blue line. And what we'll see is this notion of steepness, how steep a line is, how quickly does it increase or how quickly does it decrease, is a really useful idea in mathematics. So ideally, we'd be able to assign a number to each of these lines or to any lines that describes how steep it is, how quickly does it increase or decrease? So what's a reasonable way to do that? What's a reasonable way to assign a number to these lines that describe their steepness? Well one way to think about it, could say well, how much does a line increase in the vertical direction for a given increase in the horizontal direction? So let's write this down. So let's say if we an increase increase, in vertical, in vertical, for a given increase in horizontal for a given increase a given increase in horizontal. So, how can this give us a value? Well let's look at that magenta line again. Now let's just start at an arbituary point in that magenta line. But I'll start at a point where it's going to be easy for me to figure out what point we're at. So if we were to start right here, and if I were to increase in the horizontal direction by one. So I move one to the right. To get back on the line, how much do I have to increase in the vertical direction? Well I have to increase in the vertical direction by two. By two. So at least for this magenta line, it looks like our increase in vertical is two, whenever we have an increase in one in the horizontal direction. Let's see, does that still work if I were to start here, instead of increasing the horizontal direction by one, if I were increase in the horizontal direction... So let's increase by three. So now, I've gone plus three in the horizontal direction, then to get back on the line, how much do I have to increase in the vertical direction? I have to increase by one, two, three, four, five, six I have to increase by six. So plus six. So when I increase by three in the horizontal direction, I increase by six in the vertical. We were just saying, hey, let's just measure how much to we increase in vertical for a given increase in the horizontal? Well two over one is just two and that's the same thing as six over three. So no matter where I start on this line, no matter where I start on this line, if I take and if I increase in the horizontal direction by a given amount, I'm going to increase twice as much twice as much in the vertical direction. Twice as much in the vertical direction. So this notion of this increase in vertical divided by increase in horizontal, this is what mathematicians use to describe the steepness of lines. And this is called the slope. So this is called the slope of a line. And you're probably familiar with the notion of the word slope being used for a ski slope, and that's because a ski slope has a certain inclination. It could have a steep slope or a shallow slope. So slope is a measure for how steep something is. And the convention is, is we measure the increase in vertical for a given in increase in horizontal. So six two over one is equal to six over three is equal to two, this is equal to the slope of this magenta line. So let me write this down. So this slope right over here, the slope of that line, is going to be equal to two. And one way to interpret that, for whatever amount you increase in the horizontal direction, you're going to increase twice as much in the vertical direction. Now what about this blue line here? What would be the slope of the blue line? Well, let me rewrite another way that you'll typically see the definition of slope. And this is just the convention that mathematicians have defined for slope but it's a valuable one. What is are is our change in vertical for a given change in horizontal? And I'll introduce a new notation for you. So, change in vertical, and in this coordinate, the vertical is our Y coordinate. divided by our change in horizontal. And X is our horizontal coordinate in this coordinate plane right over here. So wait, you said change in but then you drew this triangle. Well this is the Greek letter delta. This is the Greek letter delta. And it's a math symbol used to represent change in. So that's delta, delta. And it literally means, change in Y, change in Y, divided by change in X, change in X. So if we want to find the slope of the blue line, we just have to say, well how much does Y change for a given change in X? So, the slope of the blue line. So let's see, let me do it this way. Let's just start at some point here. And let's say my X changes by two so my delta X is equal to positive two. What's my delta Y going to be? What's going to be my change in Y? Well, if I go by the right by two, to get back on the line, I'll have to increase my Y by two. So my change in Y is also going to be plus two. So the slope of this blue line, the slope of the blue line, which is change in Y over change in X. We just saw that when our change in X is positive two, our change in Y is also positive two. So our slope is two divided by two, which is equal to one. Which tells us however much we increase in X, we're going to increase the same amount in Y. We see that, we increase one in X, we increase one in Y. Increase one in X, increase one in Y. >From any point on the line, that's going to be true. You increase three in X, you're going to increase three in Y. It's actually true the other way. If you decrease one in X, you're going to decrease one in Y. If you decrease two in X, you're going to decrease two in Y. And that makes sense from the math of it as well Because if you're change in X is negative two, that's what we did right over here, our change is X is negative two, we went two back, then your change in Y is going to be negative two as well. Your change in Y is going to be negative two, and negative two divided by negative two, is positive one, which is your slope again.

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Intro to slope | Algebra (video) | Khan Academy

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Voiceover] As we start to graph lines, we might notice that they're differences between lines. For example, this pink or this magenta line here, it looks steeper than this blue line. And what we'll see is this notion of steepness, how steep a line is, how quickly does it increase or how quickly does it decrease, is a really useful idea in mathematics.

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Sal analyzes what it means for a slope to be positive or negative (spoiler: it affects the direction of the line!)

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The slope of a line is rise over run. Learn how to calculate the slope of the line in a graph by finding the change in y and the change in x.

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4.  Practice finding slope of a line.  100%?  You betcha!

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4.  Practice finding slope of a line.  100%?  You betcha!

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Find the slope of the line that goes through the ordered pairs 4 comma 2 and negative 3 comma 16. So just as a reminder, slope is defined as rise over run. Or, you could view that rise is just change in y and run is just change in x. The triangles here, that's the delta symbol. It literally means "change in." Or another way, and you might see this formula, and it tends to be really complicated. But just remember it's just these two things over here. Sometimes, slope will be specified with the variable m. And they'll say that m is the same thing-- and this is really the same thing as change in y. They'll write y2 minus y1 over x2 minus x1. And this notation tends to be kind of complicated, but all this means is, is you take the y-value of your endpoint and subtract from it the y-value of your starting point. That will essentially give you your change in y. And it says take the x-value of your endpoint and subtract from that the x-value of your starting point. And that'll give you change in x. So whatever of these work for you, let's actually figure out the slope of the line that goes through these two points. So we're starting at-- and actually, we could do it both ways. We could start at this point and go to that point and calculate the slope or we could start at this point and go to that point and calculate the slope. So let's do it both ways. So let's say that our starting point is the point 4 comma 2. And let's say that our endpoint is negative 3 comma 16. So what is the change in x over here? What is the change in x in this scenario? So we're going from 4 to negative 3. If something goes from 4 to negative 3, what was it's change? You have to go down 4 to get to 0, and then you have to go down another 3 to get to negative 3. So our change in x here is negative 7. Actually, let me write it this way. Our change in x is equal to negative 3 minus 4, which is equal to negative 7. If I'm going from 4 to negative 3, I went down by 7. Our change in x is negative 7. Let's do the same thing for the change in y. And notice, I implicitly use this formula over here. Our change in x was this value, our endpoint, our end x-value minus our starting x-value. Let's do the same thing for our change in y. Our change in y. If we're starting at 2 and we go to 16, that means we moved up 14. Or another way you could say it, you could take your ending y-value and subtract from that your starting y-value and you get 14. So what is the slope over here? Well, the slope is just change in y over change in x. So the slope over here is change in y over change in x, which is-- our change in y is 14. And our change in x is negative 7. And then if we want to simplify this, 14 divided by negative 7 is negative 2. Now, what I want to show you is, is that we could have done it the other way around. We could have made this the starting point and this the endpoint. And what we would have gotten is the negative values of each of these, but then they would've canceled out and we would still get negative 2. Let's try it out. So let's say that our start point was negative 3 comma 16. And let's say that our endpoint is the 4 comma 2. 4 comma 2. So in this situation, what is our change in x? Our change in x. If I start at negative 3 and I go to 4, that means I went up 7. Or if you want to just calculate that, you would do 4 minus negative 3. 4 minus negative 3. But needless to say, we just went up 7. And what is our change in y? Our change in y over here, or we could say our rise. If we start at 16 and we end at 2, that means we went down 14. Or you could just say 2 minus 16 is negative 14. We went down by 14. This was our run. So if you say rise over run, which is the same thing as change in y over change in x, our rise is negative 14 and our run here is 7. So notice, these are just the negatives of these values from when we swapped them. So once again, this is equal to negative 2. And let's just visualize this. Let me do a quick graph here just to show you what a downward slope would look like. So let me draw our two points. So this is my x-axis. That is my y-axis. So this point over here, 4 comma 2. So let me graph it. So we're going to go all the way up to 16. So let me save some space here. So we have 1, 2, 3, 4. It's 4 comma-- 1, 2. So 4 comma 2 is right over here. 4 comma 2. Then we have the point negative 3 comma 16. So let me draw that over here. So we have negative 1, 2, 3. And we have to go up 16. So this is 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. So it goes right over here. So this is negative 3 comma 16. Negative 3 comma 16. So the line that goes between them is going to look something like this. Try my best to draw a relatively straight line. That line will keep going. So the line will keep going. So that's my best attempt. And now notice, it's downward sloping. As you increase an x-value, the line goes down. It's going from the top left to the bottom right. As x gets bigger, y gets smaller. That's what a downward-sloping line looks like. And just to visualize our change in x's and our change in y's that we dealt with here, when we started at 4 and we ended at-- or when we started at 4 comma 2 and ended at negative 3 comma 16, that was analogous to starting here and ending over there. And we said our change in x was negative 7. We had to move back. Our run we had to move in the left direction by 7. That's why it was a negative 7. And then we had to move in the y-direction. We had to move in the y-direction positive 14. So that's why our rise was positive. So it's 14 over negative 7, or negative 2. When we did it the other way, we started at this point. We started at this point, and then ended at this point. Started at negative 3, 16 and ended at that point. So in that situation, our run was positive 7. And now we have to go down in the y-direction since we switched the starting and the endpoint. And now we had to go down negative 14. Our run is now positive 7 and our rise is now negative 14. Either way, we got the same slope.

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Find the slope of the line that goes through the ordered pairs 4 comma 2 and negative 3 comma 16. So just as a reminder, slope is defined as rise over run. Or, you could view that rise is just change in y and run is just change in x.

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6.  Practice finding slope from two points.  Yep, you guessed it.  Get 100%!

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6.  Practice finding slope from two points.  Yep, you guessed it.  Get 100%!

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Slope from two points | Algebra (practice) | Khan Academy

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Given two points on a line, you can find the slope of the line. Watch Sal doing a bunch of examples.

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8.  Slope review. Read through these notes.  Well, you got it now?  

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8.  Slope review. Read through these notes.  Well, you got it now?  

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The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

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When two points have the same y-value, it means they lie on a horizontal line. The slope of such a line is 0, and you will also find this by using the slope formula.

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When two points have the same y-value, it means they lie on a horizontal line. The slope of such a line is 0, and you will also find this by using the slope formula.

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Practice your knowledge of horizontal and vertical lines: their graphical presentation, their slopes, and their equations.

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- [Voiceover] 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.

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Intro to slope-intercept form (y=mx+b) | Algebra (video) | Khan Academy

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Voiceover] 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.

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2.  More on slope-intercept form.  Read through these notes.

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2.  More on slope-intercept form.  Read through these notes.

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Learn about the slope-intercept form of two-variable linear equations, and how to interpret it to find the slope and y-intercept of their line.

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Worked examples practicing the relationship between slope, y-intercept, and linear equation in slope-intercept form.

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Practice your basic understanding of slope-intercept form for linear equations.

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5.  Graphing lines from equations written in slope-intercept form.

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5.  Graphing lines from equations written in slope-intercept form.

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Graph from slope-intercept equation (video) | Khan Academy

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To graph a linear equation in slope-intercept form, we can use the information given by that form. For example, y=2x+3 tells us that the slope of the line is 2 and the y-intercept is at (0,3). This gives us one point the line goes through, and the direction we should continue from that point to draw the entire line.

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6.  More on graphing lines from equations written in slope-intercept form.  Read through these notes.

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6.  More on graphing lines from equations written in slope-intercept form.  Read through these notes.

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Learn how to graph lines whose equations are given in the slope-intercept form y=mx+b.

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7.  Practice graphing lines from equations written in slope-intercept form.  Get 100%!

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7.  Practice graphing lines from equations written in slope-intercept form.  Get 100%!

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Graphing slope-intercept equations

Graphing slope-intercept equations

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Graph from slope-intercept form (practice) | Khan Academy

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8.  Slope-intercept form review.  Read through.  You got this?

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8.  Slope-intercept form review.  Read through.  You got this?

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Graphing lines from slope-intercept form review

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Slope intercept form is y=mx+b, where m is slope and b is the y-intercept. We can use this form of a linear equation to draw the graph of that equation on the x-y coordinate plane.