Modeling with linear tables

0:00- [Lecturer] We're told Tierra is updating her laundromat to use a water reclamation system. The system cleans the water the washing machines use so they can use it again. That sounds useful. The laundromat has 60 washing machines. For each washing machine that Tierra connects to the system, the average daily waste water from the laundromat goes down by 5 1/3 cubic meters.

0:20Before using the system, the laundromat had an average of 400 cubic meters of wastewater per day. Complete the table to show the average volume w of wastewater from the laundromat if Tierra connects m machines to the system. So there's a couple of ways that you could approach this, and I encourage you to pause this video and try this on your own before we do this together.

0:44One way to just say, all right, I'm going to start with 400 cubic meters. And every time, let's say I had one machine, I'm adding one machine, well, that's going to take 5 1/3. And actually I like dealing with non-mixed numbers. So that's the same thing, 5 1/3 is the same thing as 16 over 3.

1:03So that's if you had one machine. If you had 202 machines, it would be minus 2 times 16/3, et cetera, et cetera. And then you do that for 9 and 20. But we can actually set up a general equation here that might make this a little bit simpler. We could just say that, and I don't have much space, so I'm utilizing it as much as I can.

1:24We could say that the average volume of wastewater w is going to be equal to the starting point, 400 cubic meters, minus 5 1/3 cubic meters. I'll just write 16 over 3. Make the math a little bit easier later on. Times the number of machines that I add to the system.

1:42And so now we can use this equation to solve for what we don't know. So for example, when m is equal to 9, we are going to get w is equal to 400 minus 16 over 3 times 9, 16 over 3 times 9, 9 divided by 3 is 3. So it's 400 minus 48, which is equal to 352. And so let me actually delete all of this 'cause I think I'm gonna need this real estate again.

2:18(keyboard types) So this is all going to be equal to 3, whoops, not using the right tool. This is going to be equal to 352. Now what happens when m is 20? We get w is equal to 400 minus 16 over 3 times 20. Well, what's that going to be? This is going to be w is equal to, lemme just write everything over 3.

2:47This is 1,200 over 3. That's the same thing as 400, and 16 over 3 times 20. Let's see, 2 times 16 is 32 times another 10, so it's minus 320 over 3. And so this is going to be equal to, let's see, 1,200 minus 300 is 900, and I'm gonna take out another 20. So that is gonna be 880 over 3.

3:17And we could write it that way if we want, or we could write it as a mixed number. So lemme just write 880 over 3. And now we're going to have to go the other way around. So now we are told the average wastewater volume, we're told w and we need to figure out the number of machines.

3:38So here we get 320 is equal to 400 minus 16 over 3 m. Well if we subtract 400 from both sides, we are going to get negative 80 is equal to negative 16 over 3m. To solve for m, we could multiply both sides times the reciprocal of the coefficient. So that's going to be negative 3 over 16.

4:02So times negative 3 over 16. On the right-hand side, we just end up with an m. On the left-hand side, the negatives cancel out. 80 divided by 16 is 5 and then times 3 is going to be 15. So in this situation, m is equal to 15. And then last but not least, and I have a feeling I'm going to have to use some of this real estate again, so lemme clear out all of this.

4:29(keyboard typing) We have a situation where w is 133 1/3. Lemme write that as, not as a mixed number. So that's going to be 399 plus 1, so this is 400 over 3. So I could write, when w is 400 over 3 is equal to, is equal to 400. And lemme write that as 1,200 over 3.

5:00Minus 16 over 3 m. While I can subtract 1,200 over 3 from both sides, and so I'm going to get negative 800 over 3 is equal to negative 16 over 3 m. And I could do something very similar where I multiply both sides times negative 3 over 16 times negative 3 over 16.

5:26On the right-hand side, negatives cancel out, 3 cancel out, 16s cancel out. I'm left with just an m. And on the left-hand side, negatives cancel out, 3s cancel out, 80 divided by 16 is 5, so 800 divided by 16 is 50. So in this situation, m is equal to 50. And we're done.