Ex 1: Distributive property to simplify

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.