# Ex 1: Distributive property toÂ simplify

## Video transcript

In this video, we're going to
a couple of warm-up solving equations problems. And you'll
see these require several steps, maybe a little bit more
than the ones that we've done in the previous video. And then we'll do a word problem
that applies our equation-solving capabilities. So here we have 7 times w plus
20, minus w, is equal to 5. Let's see if we can
solve this. And like all things, there's
multiple ways to solve these. I'll just solve the way that
seems most natural to me. So one thing I like to do, is
I like to distribute the numbers out. Because if I distribute the
numbers out, I get a 7w, and then I can subtract
w from there. So maybe I can merge the
w terms somehow. So this 7w plus 20 I can rewrite
as 7w plus 7 times 20. Remember, distributive
properties. So plus 140. And then we have this minus
w is equal to 5. I just rewrote this
part right here. I just distributed the 7. And now, just like I talked
about, we can merge-- We can take 7w, and from that,
we can subtract a w. So if you take those two
terms, you get 6w. If I have 7 of something, and
I take away 1 of that something, I have 6
of that something. So I have 6w plus 140
is equal to 5. Now, I want get rid
of this 140. Because then I'll have 6w is
equaling to something, and I'll be able divide by
6, and all of that. So to get rid of 140, I can
subtract 140 from both sides of the equation. And I'll do it in pink. Minus 140. So I'm just subtracting
140 from both sides of this equation. If something equals something,
something minus 140 is going to equal something minus 140. Whatever you do to one side,
you've got to do to the other side. So the whole point here was for
these two to cancel out. You're left with the left side,
which is 6w is equal to 5 minus 140. Well, that's negative 135. And now we can divide both sides
of this equation by 6, which is equivalent to
multiplying by 1/6, and you get w is equal to, let's see,
negative 135 over 6. Let's see, is there any place
to simplify this anymore? Let's see. This isn't divisible by 2,
and it also doesn't look divisible by 3. So this looks like we are
done with the problem. And you can verify. Actually let's verify, because
it's kind of a strange-looking-- Let's verify
that this is the answer. So 7 times negative 135 over
6-- that's our solution for w --plus 20. Instead of 20, I'm going to
write plus 120 over 6. 20 is the same thing as
120 over 6, right? Minus w. So w is negative 135 over 6. So subtracting a negative
becomes adding a positive, right? So let's see what happens. This becomes 7 times-- Let's
see, negative 135 plus 120 is negative 15 over 6
plus 135 over 6. Let's see what we get here. So then we get 7 times 15. So let me go over here. What is 7 times 15? It's 70 plus 35. So it's negative 105 over 6. That is that right there. Plus 135 over 6. What does that become? This becomes 30 over 6,
which is equal to 5. Which is exactly what it
needed to be equal to. It's equal to 5, so we
got the right answer. My spider sense was wrong. We did it correctly, even though
we got this strange looking answer. So now let's do this problem. So once again, I like to
distribute out the 9. Actually, we don't have
to distribute it out. There's multiple ways
to do this. Maybe we'll do it both ways. So the first way I like to do it
is to distribute out the 9, so I don't have to deal
with fractions. So you get 9x minus 18. just
distributed the 9. Is equal to 3x plus 3. Now we want to get the x-terms
together somehow. Let's get them together
on the left-hand side. So let's get rid of this 3x
on the right-hand side. And the best way to get rid of
it is to subtract 3x from the right-hand side. But if we do it from the
right-hand side, we have to do it from the left-hand
side, as well. So I'm subtracting 3x
from both sides. The left-hand side, 9x minus 3x
is equal to 6x, and then, of course, you have your minus
18 is equal to 3x minus 3x. That just disappears, those
cancel out, and you just have the 3 left over. Now there's multiple ways
you could do this. I mean, one fun thing-- Well,
let me just do it the most traditional way. We could add 18 to both sides so
that the 18 disappears from the left-hand side. So then you are left with 6x--
these two guys cancel out --is equal to 3 plus 18,
which is 21. Divide both sides by 6, you get
x is equal to 21 over 6, or if you divide the numerator
and the denominator by 3, you get 7 over 2. And you are done. Now, I said that there's
multiple ways to do this problem. Let me do it another
way here in orange. So you have 9 times x minus
2 is equal to 3x plus 3. Well, I see a 9,
I see some 3's. What if I just divide both sides
of this equation by 3? So if I divide that side by 3,
and I divide all of these terms by 3. What do I get? This becomes 3 times x minus
2 is equal to x plus 1. Maybe at this point, if I want,
I could distribute this. So this becomes 3x minus
6 is equal to x plus 1. I could subtract x from both
sides of this equation, so I get 2x minus 6 is equal to 1. Remember, I subtracted that
from both sides, so it disappeared on the
right-hand side. I could add 6 to both sides
of this equation. I get 2x is equal to 1
plus 6, which is 7. Divide both sides by 2, you get
x is equal to 7 over 2. I went through this a
little bit faster. But really, I just wanted to
show you that as long as do legitimate operations, you're
going to get the same answer. And you could check, verify
that this is indeed the correct answer, if you like. Now we have a word problem. Let's see if we can
tackle this. Lydia inherited a
sum of money. She split it into
5 equal chunks. So let me just say m is the
amount of money she has. So she split into
5 equal chunks. So let me say, Lydia's money
that she inherited. She splits it into
5 equal chunks. She invested 3 parts of the
money in a high interest bank. So how much did she invest in
the high interest bank? She divided it into 5 equal
chunks, and she invested 3 of those chunks into a high
interest bank account. So she took her money, divided
it into 5 chunks-- So this is each of the 5 chunks. Then she put 3 of those chunks,
or you could say she took 3/5 of her money, and she
put it into the high interest bank account, which adds
10% to the value. So that's how much she
originally put into high interest banking account. She placed the rest of her
inheritance plus $500 in the stock market. So how much was that? So she placed the rest
of her inheritance. So she put 3/5 of it in the high
interest bank account. What's left over? What's going to be the 2/5? 2/5 of her money she is
going to invest in the stock market, right? You combine 3/5 plus 2/5,
you have all of the money that she inherited. But she didn't put
just the 2/5. She put the rest of her
inheritance, which is the 2/5 m, plus $500, in the stock
market, but lost 20%. So this is how much she put in,
and this is how much she ends up with. So put in is right there,
and then ends up with. So on the checking accounts,
it added 10% of its value. So she started with 3/5 of her
money, and it added another 10% of that. So plus, let's say, 0.10
times the amount of money she put in. Times 3/5 of her money. This is how much she
ends up with. Her original amount that she put
into the account plus 10% of the original amount. It grew by 10%. It added 10% of the value. Now in the stock market, she
started with 2/5 m plus 500, but she lost 20%
on that money. So she loses 0.20 for 20%,
times 2/5 m plus 500. That's how much she loses
on the market. She loses 20% of this
amount of money. Now at the end it says, if the
two accounts end up with the exact same amount of
money in them, how much did she inherit? So this and this are going to
end up being equal, and we'll have to solve for m. So let's do that. We get 3/5 m plus-- Well, let's
see, this is the same thing as 1/10, right? Let me write that way. So 1/10 times 3/5 is 3/50. Plus 3/50 m is equal to-- I just
multiplied the 1/10 times 3/5 --is equal to 2/5 m-- I
want to do it in that same color, in the orange --is
equal to 2/5 m plus 500. And then 0.2 is the same
thing as 1/5, right? 0.2-- let me write it over here
--is equal to 20/100, which is equal to 1/5. So we can rewrite this
right here as 1/5. So 2/5 m plus 500 minus
1/5 times 2/5 plus 5 2/5 m plus 500. So that's a hairy problem, but
well take it step by step and see that it's not so bad. So right here, let's add
3/5 of something plus 3/50 of something. Well, 3/5 is the same thing
as 30/50, right? If I multiply the numerator
and the denominator by 10. And now we can add this. 30/50 plus 3/50 is 33/50 m is
equal to-- and let's just simplify this a little
bit --2/5 m plus 500. Distribute the negative 1/5,
so you get negative 2/25 m, and then negative 1/5 times
500 is minus 100. Let's simplify this even more. The left-hand side is still 33
over 50 m is equal to-- And now we have we have these
coefficients on our m terms, right here. Those are our m terms. So you could view it as
2/5 minus 2/25 m. that takes care of that
term and that term. I just factored the m out. And then you have the
500 minus 100. So that's plus 400. Now, let's see. 2/5, if we multiplied the
numerator and denominator by 5, this becomes 10/25. Right? So our whole equation is now
33/50 m is equal to-- what is 10 minus 2? So that's 8/25 m plus 400. We're getting close! We're getting close. Now let's get both
m terms onto the left-hand side of the equation. So let's subtract 8/25
m from both sides. Did that, so that the right-hand
side cancels outs. So our right-hand side
is just equal to 400. And then our left-hand side
is 33/50 minus 8 over 25. So it's equal to 33 over
50 minus 8 over 25. That's the same thing as minus
16 over 50, right? I just multiply the numerator
and denominator by 2. m is equal to 400. We're almost there! This is a nice, meaty problem. Almost there. And then 33 minus
16 is 17, right? So we're left with 17/50
m is equal to 400. And now we can multiply
both sides times the inverse of 17/50. So 50/17 times 50/17. These cancel out, and you get
m is equal to-- and I'll get the calculator out for this. m is equal to 400 times
50 divided by 17 is equal to $1,176.47. That's how much Lydia
started out with. Hopefully you found that fun.