用點斜式呈現的線性方程式 (英)

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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.