Two-step equation word problems
Now that we've set up our equations, let's see if we can answer some more interesting questions. Just as a warm-up, we're going to do a little bit of what we did in the last video. But I'll do it with the equations now. The equations really are just a mathematical representation of the same information that we had up here. So as a starting point, let's think about how much, under the basic plan, it'll cost for us to attend 3 sessions in a given month. Well, we've got our equation right over here. Our total cost in that month is going to be equal to 20 plus 8 times the number of sessions. The number of sessions is S. S is 3 in this circumstances, so we'll replace the S with the 3. And so you get our cost in this circumstance is going to be 20 plus 8 times 3, or 20 plus 24, or $44. Let me erase this right over here. So this is going to cost us $44. Now let's do the same thing under the trial plan. How much is it going to cost us to attend 3 sessions? Well, this is more straightforward, or slightly more straightforward. Our total cost is going to be equal to 12 times the number of sessions. The number of sessions in this circumstance are 3 sessions, so it's going to be 12 times 3. Our total cost is going to be equal to $36. Let me write that down. It's going to be equal to $36. But we could also answer more interesting questions with these equations. For example, how many sessions can I attend in a month for, oh, I don't know, let's throw out something, $54. Let's say that's just my budget. When I look at my salary and how much money I have budgeted to health and fitness, this is what I can afford. And so based on that, I want to ask, how many sessions can I attend in each of these situations? First, let's think about it for the trial plan. How many sessions can I attend for $54? Now let's just assume that this is some type of a special yoga studio where you can even attend half a session. So if you attend half a session, they'll charge you $6. If you attend a fourth of a session, they'll charge you $3. So they'll actually keep track of exactly what fraction of a session you're attending. You don't have to just attend whole number of sessions. You could attend 2 and 1/2, and they'll bill you accordingly. With that said, how many sessions under the trial plan can I attend for $54? Well, let's think about this a little bit. My cost now is going to be $54. $54 is C. It's my monthly cost. So 54 is going to be equal to $12 per session times the number of sessions. Now I set up this equation. In order to answer this question, I just need to figure out what's the number of sessions that satisfy it. Now this S is an unknown variable. What number of sessions times 12 is equal to 54? Well, you might say, well, if I could get just a S on the right-hand side here, then I'll have an answer. S will be equal to some value. And if I want just a S on the right-hand side, I'd ideally just like to divide this by 12. 12S divided by 12 is just going to give me a S. It's just going to give me the number of sessions. But if I have an equal sign, I can't just willy-nilly divide one side by 12 without doing the exact same thing to the other side. If that is equal to that, in order for them to both still be equal, I have to do the same thing to both sides. So on the right-hand side, I'm left with just a S. And on the left-hand side, I have 54 divided by 12. Let me think about that. I'll just work it out. 12 goes into 54 4 times. 4 times 12 is 48. You're left with a remainder of 6. So you could say 12 goes into 54, or you could say 54/12 is equal to 4 and 6/12 or 4 and 1/2. So the number of sessions that I could attend are 4 and 1/2. If I have $54 to spend, I could attend 4 and 1/2 sessions under the trial plan. If this was a yoga studio that said, oh, you can't attend half sessions, then you could say, well, at most I could attend four sessions, because five sessions would get me above my budget. But I said that you can attend fractional sessions, so we're going to say 4 and 1/2. Now let's answer the same question for the basic plan. Under the basic plan, how many sessions can I attend for $54? Well, once again, this is going to be our total cost for that month. So we say $54 is equal to 20 plus 8 times the number of sessions. And once again, I'd like to just have sessions maybe on the right-hand side of this equation. So I could say, oh, sessions are going to be equal to something. This is the number of sessions that I could attend. But I have this 20 here and this 8 here, and so I have to think about how I can get rid of them. The first thing I've got to think about is, how can I get rid of this 20 from the right-hand side of this equation? What am I going to have to do to the entire equation? Well, the simplest thing is I could subtract 20 from the right-hand side. But if I subtract 20 from the right-hand side, I also have to subtract 20 from the left-hand side in order to maintain the equality. So I'm left with, on the left-hand side, 54 minus 20 gets me 34. And that's going to be equal to-- well, the whole point of subtracting 20 is so that these negate each other. And then I have just an 8S. So 8 times the number of sessions has to be equal to 34 in order for 8 times the number of sessions plus 20 to be equal to 54. And then you could add 20 to both sides of this equation to get our original. We're almost done. We just want to isolate the S right here. And you can imagine the easiest way would be to divide the right-hand side by 8. But if we're dividing the right-hand side by 8, we have to do the same to the left. And so what do we get for the number of sessions that we would have to attend? Well, these cancel out. And so you get the number of sessions is equal to 34 divied by 8. 34 divided by 8 is equal to, let's see, 8 goes into 34 four times with a remainder of 2, so 4 and 2/8, or it's 4 and 1/4. So I could attend 4 and 1/4 sessions under the basic plan. Assuming I can do fractional sessions, for $54, which one can I attend more sessions in, the basic plan or the trial plan? Well, for $54 I could attend a slightly longer amount of sessions under the trial plan.