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Intro to point-slope form

Point-slope is the general form y-y₁=m(x-x₁) for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept). Watch this video to learn more about it and see some examples. Created by Sal Khan.

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  • blobby green style avatar for user c1.Malik
    Wait then what form is y = mx + b
    (2 votes)
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  • aqualine ultimate style avatar for user plewis283
    At about seconds or so. He says that those triangles are the deltas. I understand that but for full formula for slope does it matter which y or x goes first? Would you still get the same answer?
    (7 votes)
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    • leaf orange style avatar for user Benny C
      Slope is always rise over run. It doesn't matter which one you find first, but make sure they're in the proper place.

      Consider a line with rise 5 and run 4. The rise/run way is 5/4. But the run/rise way is 4/5. That is a different value, and would give us with a completely different line.
      (21 votes)
  • blobby green style avatar for user paulacallorarecto
    I am a student teacher and I have difficulty in thinking about an activity that will lead to this subject. They want it to be a discovery activity that will also serve as a motivational activity for this lesson. I refered to books, but there's no discovery activity for this lesson. Do you have anything in mind?
    (4 votes)
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    • duskpin ultimate style avatar for user Pei-Hsin Lin 林培心 🌸
      Hi, Paula. Here are some ideas:

      1. One way to think about point-slope form is as a rearrangement of the slope formula.
      If you ask your kids to manipulate m = (y - k)/(x - h), perhaps one will come up with (y - k) = m(x - h).

      2. Another way to think about point-slope form is as a transformation of the canonical line y = mx: That is to say, (y - k) = m(x - h) is the end result of a vertical translation by k units, and a horizontal translation by h units, performed in either order.

      3. Also food for thought: Given a point (h,k) and a slope m, the equation
      (y - k) = m(x - h) is guaranteed to evaluate as 0 = m·0 = 0.
      Of course 0 is the product of any number and 0. This conceptually echoes how polynomial factors yield roots, based on the fact that any zero product must have one or more zero factors (aka the Zero Product Property).
      (6 votes)
  • female robot grace style avatar for user Rey #FilmmakerForLife #EstelioVeleth.
    what is the traditional point-slope formula?
    (6 votes)
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  • primosaur seedling style avatar for user Jazzy T
    My math teacher uses an equation of y-y1 = m(x-x1). Is this equation equal to the one in the video? If so, what would the (a,b) be taking the place of?
    (5 votes)
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  • starky sapling style avatar for user Nithin
    I don't understand pointform at all please explain someone
    (4 votes)
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    • starky sapling style avatar for user Sanya Singh
      The point-slope form is very useful when you don't have your y-intercept. It is used to write equations when you only have your slope and a point.
      Point-slope form: y-a = m(x-b).
      For example, your slope (m) is 3 and your point (a,b) is 9,10.
      You would substitute your y-coordinate for a, and your x- coordinate for b. Your new equation would look like this: y-10 = 3(x-9). If you simplify this, then you will get your basic slope-intercept form: y=mx+b! I hope this made sense!
      (5 votes)
  • blobby green style avatar for user 22asmith
    what would i replace M with
    (3 votes)
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  • duskpin ultimate style avatar for user Jinsung Lim
    Why is slope referred by 'm'? Is it some kind of short form?
    (2 votes)
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    • male robot donald style avatar for user Saadali2000
      It is not known why the letter m was chosen for slope; the choice may have been arbitrary. John Conway has suggested m could stand for "modulus of slope." One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for "to climb" is monter. The earliest known use of m for slope is an 1844 British text by Matthew O'Brien entitled "A Treatise on Plane Co-Ordinate Geometry"
      (3 votes)
  • duskpin seedling style avatar for user Kittypaw
    This is so confusing!! I just don't understand.
    (3 votes)
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  • aqualine tree style avatar for user C
    How do I know which pair of co-ordinates to use as the starting point?
    For example, if these were the two points (5,1) and (6, 2)
    would I do 1-2 over 5-6 or would I do 2-1 over 6-5?
    Does it matter? I'm very confused.
    (3 votes)
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Video transcript

So what I've drawn here in yellow is a line. And let's say we know two things about this line. We know that it has a slope of m, and we know that the point a, b is on this line. And so the question that we're going to try to answer is, can we easily come up with an equation for this line using this information? Well, let's try it out. So any point on this line, or any x, y on this line, would have to satisfy the condition that the slope between that point-- so let's say that this is some point x, y. It's an arbitrary point on the line-- the fact that it's on the line tells us that the slope between a, b and x, y must be equal to m. So let's use that knowledge to actually construct an equation. So what is the slope between a, b and x, y? Well, our change in y-- remember slope is just change in y over change in x. Let me write that. Slope is equal to change in y over change in x. This little triangle character, that's the Greek letter Delta, shorthand for change in. Our change in y-- well let's see. If we start at y is equal to b, and if we end up at y equals this arbitrary y right over here, this change in y right over here is going to be y minus b. Let me write it in those same colors. So this is going to be y minus my little orange b. And that's going to be over our change in x. And the exact same logic-- we start at x equals a. We finish at x equals this arbitrary x, whatever x we happen to be at. So that change in x is going to be that ending point minus our starting point-- minus a. And we know this is the slope between these two points. That's the slope between any two points on this line. And that's going to be equal to m. So this is going to be equal to m. And so what we've already done here is actually create an equation that describes this line. It might not be in any form that you're used to seeing, but this is an equation that describes any x, y that satisfies this equation right over here will be on the line because any x, y that satisfies this, the slope between that x, y and this point right over here, between the point a, b, is going to be equal to m. So let's actually now convert this into forms that we might recognize more easily. So let me paste that. So to simplify this expression a little bit, or at least to get rid of the x minus a in the denominator, let's multiply both sides by x minus a. So if we multiply both sides by x minus a-- so x minus a on the left-hand side and x minus a on the right. Let me put some parentheses around it. So we're going to multiply both sides by x minus a. The whole point of that is you have x minus a divided by x minus a, which is just going to be equal to 1. And then on the right-hand side, you just have m times x minus a. So this whole thing has simplified to y minus b is equal to m times x minus a. And right here, this is a form that people, that mathematicians, have categorized as point-slope form. So this right over here is the point-slope form of the equation that describes this line. Now, why is it called point-slope form? Well, it's very easy to inspect this and say, OK. Well look, this is the slope of the line in green. That's the slope of the line. And I can put the two points in. If the point a, b is on this line, I'll have the slope times x minus a is equal to y minus b. Now, let's see why this is useful or why people like to use this type of thing. Let's not use just a, b and a slope of m anymore. Let's make this a little bit more concrete. Let's say that someone tells you that I'm dealing with some line where the slope is equal to 2, and let's say it goes through the point negative 7, 5. So very quickly, you could use this information and your knowledge of point-slope form to write this in this form. You would just say, well, an equation that contains this point and has this slope would be y minus b, which is 5-- y minus the y-coordinate of the point that this line contains-- is equal to my slope times x minus the x-coordinate that this line contains. So x minus negative 7. And just like that, we have written an equation that has a slope of 2 and that contains this point right over here. And if we don't like the x minus negative 7 right over here, we could obviously rewrite that as x plus 7. But this is kind of the purest point-slope form. If you want to simplify it a little bit, you could write it as y minus 5 is equal to 2 times x plus 7. And if you want to see that this is just one way of expressing the equation of this line-- there are many others, and the one that we're most familiar with is y-intercept form-- this can easily be converted to y-intercept form. To do that, we just have to distribute this 2. So we get y minus 5 is equal to 2 times x plus 2 times 7, so that's equal to 14. And then we can get rid of this negative 5 on the left by adding 5 to both sides of this equation. And then we are left with, on the left-hand side, y and, on the right-hand side, 2x plus 19. So this right over here is slope-intercept form. You have your slope and your y-intercept. So this is slope-intercept form. And this right up here is point-slope form.