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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.

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