線性方程式的標準式 (英)

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線性方程式的標準式 (英) : 線性方程式的標準式

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Let's do some examples dealing with equations of lines in
standard form.
So, so far we've had two other forms. We've had
slope-intercept, which is of the form, y is
equal to mx plus b.
That's actually this right here.
This is in slope-intercept form.
We've seen point-slope form in the last video.
That's of the form, y minus some y-value on the line being
equal to the slope times x minus some x-value on the
line, when you have that y-value.
So the point x1, y1 is on the line.
This right here is an example of point-slope form.
And now we're going to talk about the standard form.
And the standard form-- let me write it here-- standard form
is essentially putting all of the x and y terms onto the
left-hand side of the equation.
So you get ax plus by is equal to c.
I want to really emphasize that all of these are just
different ways of writing the same equation.
If you're given this, you can out algebraically manipulate
it to get to that or to that.
If you're given that, you can get to that or that.
These are all different ways of writing the exact same
relationship, the exact same line.
So let's do a couple of examples of this.
So here we have a line right here.
We have an equation written in slope-intercept form.
The slope is 3, the y-intercept is negative 8.
Let's put it into standard form.
So we just have to get the 3x onto the
other side of the equation.
And the best way I can think of doing that-- let me rewrite
the equation, y is equal to 3x minus 8-- let's some subtract
3x from both sides of the equation.
So if you subtract 3x from both sides-- so you subtract
3x, subtract 3x-- what do the left- and right-hand sides of
the equation become?
The left-hand side becomes negative 3x plus y being equal
to-- the 3x and the negative 3x cancel out-- being equal to
negative 8.
We're done.
That's standard form right there.
Standard form, I guess people like it because it has both
the coefficients on the left-hand side.
But it's kind of useless in trying to figure out slope and
y-intercept.
I don't know what the slope and y-intercept is when I look
at it in standard form.
My favorite is slope-intercept form, because it tells you
exactly the slope and an intercept.
Point-slope, easy to get to, and you can look at it and
figure out the slope.
But y-intercept, you have to do a little bit of work to
figure it out.
But at least you can just go immediately from the slope and
a point to it.
But anyway, let's go from this equation, which is written in
point-slope form, and get it to the standard form.
So we want to get it to the standard form, to the same
type of standard form.
So a good thing to do, let's just distribute things out. y
minus 7 is equal to negative 5 times x, negative 5x, plus
negative 5, times negative 12, which is positive 60.
Now, we want all of the variable terms on the left,
all of the constant terms on the right.
So let's add 7 to both sides of this equation.
So plus 7 to both sides of this equation.
What does it become?
Well, the minus 7 disappears, because negative 7 plus 7.
So you're just left with a y being equal to
negative 5x plus 67.
Now, if we want this x term on the left-hand side, we could
add 5x to both sides.
So let's add 5x to both sides of this equation.
And we will get y plus 5x is equal to-- these
cancel out-- 67.
Now, this is pretty much standard form.
If you really want to be a stickler for it, you can
rearrange these two.
So it'd be 5x plus y is equal to 67.
And you are done.
Let's do one more of these.
So this is in neither point-slope nor in
slope-intercept form.
It's just in some type of intermediary
mixed form right there.
This looks like some type of point-slope, but this looks
like something different.
So it's really not point-slope.
Let's see if we can algebraically manipulate it to
the standard form.
So we get 3y plus 5.
Let's distribute out this 4.
So it's equal to 4x minus 36.
Let's do exactly what we did in the last.
I'm using different notation on purpose, to expose you to
different things.
So instead of doing it this way, I'm going to subtract 5
from both sides, but I'm going to do it on the same line.
So I'm going to subtract 5 from both sides.
And so the left-hand side of this equation becomes 3y,
because these two guys cancel out, and that is equal to 4x.
And then what is minus 36 minus 5?
That's minus 41.
And now we want the x terms of the left-hand side.
So let's subtract 4x from both sides of this equation.
So negative 4x plus, and then minus 4x.
What does our equation become?
Well, the left-hand side just stays negative 4x plus 3y.
And the right-hand, the reason why we subtracted 4x is so it
cancels out with that.
You just have a negative 41.
And we're done.
We are in standard form.
Now, let's go the other way.
Let's start with some equations in standard form and
figure out their slope and y-intercept.
And the best way I know to figure out the slope and
y-intercept is to put it into slope-intercept form.
So we want to put these equations right here into the
form, y is equal to mx plus b.
So we're essentially solving for y.
Let's do that.
So the best thing to do here-- so let me rewrite it.
5x minus 2y is equal to 15.
Let's subtract 5x from both sides.
So minus 5x plus, you have a minus 5x.
These cancel out.
And so you're left with negative 2y is
equal to 15 minus 5x.
And now, let's divide everything by negative 2.
If you divide everything by negative 2, what do we get?
The left-hand side just becomes a y.
y is equal to-- 15 divided by negative 2 is negative 7.5.
And then negative 5 divided by negative 2-- you can imagine
I'm distributing the negative 1/2 if you will.
I'm dividing both of these by negative 2.
So negative 5 divided by negative 2 is positive 2.5x.
And if you really wanted to put it in the slope-intercept
form, you could say that y is equal to-- you could just
rearrange these-- 2.5x minus 7.5.
You want the slope.
It's right here.
That is our slope.
You want the y-intercept.
Actually, let me be careful.
It is right there.
It is negative 7.5.
That is the y-intercept.
And now this would be a form that's actually pretty
straightforward to graph it in.
Let's do this one.
So once again, we just need to solve for y.
So let's subtract 3x from both sides.
So you get 6y is equal to 25 minus 3x.
And then you can divide both sides by 6.
So you're left with y is equal to 25 over 6 minus 3 over 6,
or minus 1/2x.
If you really want it in this from, you just rearrange this.
y is equal to negative 1/2x plus 25 over 6.
Where is the slope?
Here is the slope.
Negative 1/2, that is the slope.
Where is the y-intercept?
That's the y-intercept.
That is our b.
The point 0, 25 over 6 is on the line.
Let's do one more of these.
So we get 9x minus 9y is equal to 4.
Just for fun, let's just start off by dividing both sides of
the equation by 9.
You don't have to do it that way.
But this is kind of a fun way to do it, because the
coefficients here will immediately become 1.
So if you divide both sides of the equation by 9, if you
divide everything by 9, it becomes-- actually, well,
let's divide everything.
Let's divide everything by negative 9, even better.
I'm just doing this for fun.
So this first term will become negative x.
The second term, you have a negative 9 divided by a
negative 9, it will be a plus y.
And then this last term will just become a
negative 4 over 9.
Actually, let me write this out here.
Negative 4 over 9.
I'm giving some space there.
Now, we want the x on the right-hand side, so let's add
x to both sides of this equation.
These cancel out.
And then the equation becomes y is equal to x minus 4/9.
Where is the slope?
The slope is the coefficient on the x term.
The slope is equal to 1.
Where is the y-intercept?
The y-intercept is right there.
It is negative 4/9.