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# 多步驟方程式 (英): 多步驟方程式

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- 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.

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