利用分配律解方程式 (英)

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利用分配律解方程式 (英): 利用分配律解方程式

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We have the equation negative 9 minus this whole expression,
9x minus 6-- this whole thing is being subtracted from
negative 9-- is equal to 3 times this whole
expression, 4x plus 6.
Now, a good place to start is to just get rid of these
parentheses.
And the best way to get rid of these parentheses is to kind
of multiply them out.
This has a negative 1-- you just see a minus here, but
it's just really the same thing as having a negative 1--
times this quantity.
And here you have a 3 times this quantity.
So let's multiply it out using the distributive property.
So the left-hand side of our equation, we have
our negative 9.
And then we want to multiply the negative 1 times each of
these terms. So negative 1 times 9x is negative 9x, and
then negative 1 times negative 6 is plus 6, or positive 6.
And then that is going to be equal to-- let's distribute
the 3-- 3 times 4x is 12x.
And then 3 times 6 is 18.
Now what we want to do, let's combine our constant
terms, if we can.
We have a negative 9 and a 6 here, on this side, we've
combined all of our like terms. We can't combine a 12x
and an 18, so let's combine this.
So let's combine the negative 9 and the 6, our two constant
terms on the left-hand side of the equation.
So we're going to have this negative 9x.
So we're going to have negative 9x plus-- let's see,
we have a negative 9 and then plus 6-- so negative 9 plus 6
is negative 3.
So we're going to have a negative 9x, and then we have
a negative 3, so minus 3 right here.
That's the negative 9 plus the 6, and that is
equal to 12x plus 18.
Now, we want to group all the x terms on one side of the
equation, and all of the constant terms-- the negative
3 and the positive 18 on the other side-- I like to always
have my x terms on the left-hand side, if I can.
You don't have to have them on the left, so let's do that.
So if I want all my x terms on the left, I have to get rid of
this 12x from the right.
And the best way to do that is to subtract 12x from both
sides of the equation.
So let me subtract 12x from the right, and subtract 12x
from the left.
Now, on the left-hand side, I have negative 9x minus 12x.
So negative 9 minus 12, that's negative 21.
Negative 21x minus 3 is equal to-- 12x minus 12x, well,
that's just nothing.
That's 0.
So I could write a 0 here, but I don't
have to write anything.
That was the whole point of subtracting the 12x from the
left-hand side.
And that is going to be equal to-- so on the right-hand
side, we just are left with an 18.
We are just left with that 18 here.
These guys canceled out.
Now, let's get rid of this negative 3 from
the left-hand side.
So on the left-hand side, we only have x terms, and on the
right-hand side, we only have constant terms. So the best
way to cancel out a negative 3 is to add 3.
So it cancels out to 0.
So we're going to add 3 to the left, let's
add 3 to the right.
And we get-- the left-hand side of the equation, we have
the negative 21x, no other x term to add or subtract here,
so we have negative 21x.
The negative 3 and the plus 3, or the positive 3, cancel
out-- that was the whole point-- equals--
what's 18 plus 3?
18 plus 3 is 21.
So now we have negative 21x is equal to 21.
And we want to solve for x.
So if you have something times x, and you just want it to be
an x, let's divide by that something.
And in this case, that something is negative 21.
So let's divide both sides of this equation by negative 21.
Divide both sides by negative 21.
The left-hand side, negative 21 divided by negative 21,
you're just left with an x.
That was the whole point behind
dividing by negative 21.
And we get x is equal to-- what's 21
divided by negative 21?
Well, that's just negative 1.
Right?
You have the positive version divided by the negative
version of itself, so it's just negative 1.
So that is our answer.
Now let's verify that this actually works for that
original equation.
So let's substitute negative 1 into that original equation.
So we have negative 9-- I'll do it over here; I'll do it in
a different color than we've been using-- we have negative
9 minus-- that 1 wasn't there originally, it's there
implicitly-- minus 9 times negative 1.
9 times-- I'll put negative 1 in parentheses-- minus 6 is
equal to-- well, actually, let me just solve for the
left-hand side when I substitute a negative 1 there.
So the left-hand side becomes negative 9, minus 9 times
negative 1 is negative 9, minus 6.
And so this is negative 9 minus-- in parentheses--
negative 9 minus 6 is negative 15.
So this is equal to negative 15.
And so we get negative 9-- let me make sure I did that--
negative 9 minus 6, yep, negative 15.
So negative 9 minus negative 15, that's the same thing as
negative 9 plus 15, which is 6.
So that's what we get on the left-hand side of the equation
when we substitute x is equal to negative 1.
We get that it equals 6.
So let's see what happens when we substitute negative 1 to
the right-hand side of the equation.
I'll do it in green.
We get 3 times 4 times negative 1 plus 6.
So that is 3 times negative 4 plus 6.
Negative 4 plus 6 is 2.
So it's 3 times 2, which is also 6.
So when x is equal to negative 1, you substitute here, the
left-hand side becomes 6, and the right-hand side becomes 6.
So this definitely works out.

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