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# 用點斜式呈現的線性方程式 (英): 用點斜式呈現的線性方程式

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- Let's do some equations of lines in point-slope form.
- And this is different from y or from slope-intercept form.
- But they really are just two different ways of writing the
- same equation.
- We'll see that with a couple of examples.
- And you might remember that slope-intercept form were
- equations of the form, y is equal to mx plus b-- we did
- this in the last video-- where m is the slope, b is the
- intercept, the y-intercept.
- That's why it's called slope-intercept.
- You have the slope and the intercept.
- In point-slope form, it takes the form y minus y1-- and I'll
- tell you what y1 is in a second-- is equal to m times x
- minus x1, where the coordinate x1, y1 is a point on the line.
- That's why this is called point-slope form.
- Now, these are just two different ways of writing the
- exact same thing.
- You can always algebraically manipulate this to get that,
- or algebraically manipulate that to get that.
- And I'll show you that with a couple of examples.
- But let's just do a few in point-slope form, just to make
- things concrete in your head.
- So here we have a line that has a slope of negative 1 over
- 10, so m is equal to negative 1 over 10.
- And it goes through the point 10 comma 2.
- So we can directly go to point-slope form.
- So let's get a point.
- A point here is the point 10 comma 2.
- So we can immediately go to point-slope form. y minus,
- this is a y-value that's on the line.
- So y minus 2 is going to be equal to the slope, negative 1
- over 10 times x minus an x-value, times x minus 10,
- just like that.
- And we're done.
- We just put it in point-slope form.
- There's two things I want to point out to you.
- One, why this makes sense.
- And I also want to show you that this is
- equivalent to that.
- So the first thing is why does this make sense?
- Well, all this is saying, if you divide both sides of this
- equation by that right over there, you get y2 over x minus
- 10 is equal to negative 1 over 10.
- Or you get the change in y between any point and 2, and
- the point 2, over the change in x.
- So you get the change in y over the change in x for any
- point x, y on that line, relative to the point 10 comma
- 2, is going to be negative 1 over 10.
- This is just the definition of your slope.
- Hopefully I don't confuse you there.
- I'm just showing you that this is just using the definition
- of the slope to create the equation of the line.
- Now, the other thing I want to show is that this is
- completely equivalent to this.
- We can just algebraically manipulate this to get that.
- And let's do that.
- So this right here is the answer to the problem.
- But let's play around with it algebraically to
- get it in that form.
- So if we get y minus 2 is equal to-- let's distribute
- the negative 1 over 10.
- So it's negative 1 over 10 x, and then negative 1 over 10
- times negative 10 is plus 1.
- Now we can add 2 to both sides of this equation.
- And you get y is equal to negative 1 over 10 x plus 3.
- So just algebraically manipulating it, we were able
- to put it into slope-intercept form.
- So these two things are completely equivalent.
- Let's do a couple more problems. The line contains
- the points 10 comma 12 and 5, 25.
- So let's figure out the slope.
- So the slope, which is equal to change in y over change in
- x, is equal to-- well, let's just use this point first. So
- let's say it's 12 minus 25 over 10 minus 5.
- This is going to be equal to-- 12 minus 25 is negative 13,
- over 10 minus 5, which is 5.
- So the slope here is negative 13 over 5.
- Let's put it in point-slope form, the equation.
- So it's going to be y minus-- let's use this point right
- here-- y minus 25 is equal to the slope negative 13 over 5
- times x minus this point, 5.
- We just knew that the point 5 comma 25 is on the line.
- So y minus 25 is equal to the slope, which we figured out,
- times x minus 5.
- And we're done.
- That's all.
- If you want, out of interest, you could do the algebra to
- put this into the slope-intercept form, to the
- mx plus b form, and see that they are completely
- equivalent.
- Let's do another one.
- So they gave us our slope.
- It's 3/5, and the y-intercept is negative 3.
- So here, immediately, this would be very easy to put it
- in the slope-intercept form.
- The equation of this line is y is equal to the slope 3/5x,
- plus the y-intercept, minus 3.
- And we'd be done.
- But how do you put this into point-slope form?
- Well, we know the slope.
- We know that m is equal to 3/5.
- But do we know any points on this line?
- You need a point and a slope to immediately put it into
- point-slope form.
- Well, we know one point.
- We know the y-intercept.
- The y-intercept is negative 3.
- That means when x is equal to 0, y is equal to negative 3.
- So our point is, the point 0, negative 3.
- You could have tried to figure out other points.
- You could have said, oh, when x is equal to 5,
- y is equal to 0.
- There's all sorts of things you could've tried out.
- But this one was just sitting there for us.
- The y-intercept is negative 3.
- That means that the point 0, minus 3 is on the line.
- So let's write it in point-slope form.
- y minus this y-value, so y minus negative 3, is equal to
- the slope 3/5 times x minus this x-value, this
- x-coordinate, x minus 0.
- And this is point-slope form.
- This could be y plus 3 is equal to 3/5 times-- we could
- write times x minus 0.
- If you really wanted to make it look like point-slope form,
- this would be point-slope form, but it's kind of silly
- to write x minus 0.
- So you could just write y plus 3 is equal to 3/5x.
- It's not 100% clear that you're in point-slope
- form yet right now.
- But I think you still need to write x minus 0.
- And obviously, to go from here to there, you just have to
- subtract 3 from both sides and you'll get that.
- So these are almost equivalent.
- I mean, they are equivalent in terms of what they represent.
- They're even almost equivalent in how you write them.
- You just have to subtract 3 from both sides of this
- equation to get to that one.
- Let's do another one.
- Let's put this in point-slope form.
- They're giving us information.
- They're giving it in the form of x, f of x.
- So in this situation, when x is negative 7, f of
- x is equal to 5.
- And this coordinate, they're telling us when x is equal to
- 3, f of x is equal to negative 4.
- So just like that, we can figure out the slope first. So
- the slope, which is equal change in y over change in x.
- So let's do this.
- It's negative 4 minus 5, over 3 minus negative 7.
- And this is going to be equal to-- negative 4 minus 5 is
- negative 9.
- And then 3 minus negative 7, that's the same thing as 3
- plus 7, that's 10.
- So slope is negative 9/10.
- And so to put this in point-slope
- form is really easy.
- It's just going to be y-- let's do it this way-- y
- minus-- I'll color code it-- 3 is equal to-- I'll do this
- back to the green color-- is equal to the slope, is equal
- to negative 9 over 10, times this x minus this coordinate,
- x minus negative 4.
- Then we can close it.
- So this is in point-slope form.
- We obviously can simplify this negative a little bit.
- We can rewrite it as y minus 3 is equal to negative 9/10
- times x, plus 4.
- And we are done.

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