Heart of algebra
Interpreting linear functions — Harder example
- [Instructor] Alice fills up the gas tank of her car before going for a long drive. The equation below, this should be above, the equation above models the amount of gas, g, in gallons, in Alice's car when she has driven m miles. What is the meaning of 32 in the equation? Alright, let's look up here. This is the gallons that she has left in her car, and what's going on here? So it makes sense that, look, when m is zero, when she's driven nothing, she's going to have 15 gallons in her car. And then as you drive more and more, you're going to subtract more and more gallons from her car. So, this is, this term right over here, this is the number of gallons, this is the number of gallons used up. So why are you dividing, if you have m miles divided by 32, how do you get gallons? What are the units on the 32 need to be? And so let's just think about this. If I have miles, so this is miles up here, and I'm gonna be dividing them by some mystery units, some mystery units, I need to get gallons. I need to get gallons because this is gonna be the number of gallons used up. So what are the mystery units going to be? Well, we can treat these units the way we would treat algebraic variables. So we could solve for units here. So, we could take the reciprocal of both sides, and we could say the units, the mystery units, divided by miles, is going to be one over gallons. Or I guess you say per gallon, it's gonna be one over gallons, and then multiple both sides times miles, multiply both sides times miles, you're going to get the mystery units. The mystery units are going to be, if you multiply both sides by miles. I don't wanna get confused with this over here. So we multiply both sides by miles, both sides by miles, your mystery units are going to be miles, miles per gallon. So, this right over here, the units over here, in order for this thing to come out to miles, in order to come up to gallons, the numerator, you have m miles divided by 32 miles per gallon, miles per gallon. This is telling you how much mileage you get per gallon for her car, how much she is getting. So, let's look at the choices. Alice uses 32 gallons of gas per mile. Now we have to be very careful. It's miles per gallon, not gas per mile. Alice's tank can hold 32 gallons of gas. No, Alice's tank can hold 15 gallons of gas. She fills up her tank, and before she's driven anything, we see here, when m is zero, she's gonna have 15 gallons in her car. So, that's not right. Alice can drive 32 miles on a tank of gas. That's right. The units here are 32 miles per gallon. So, that's that one. Alice's car can drive 32 miles to the gallon. Oh, that was a close one. This isn't 32 miles per gallon, this is 32 miles on a tank of gas. This would be 32 miles over 15 gallons. No, that's not right. That was a close one. Alice's car can drive 32 miles per gallon. That's right. 32 miles per gallon. And if this unit business that I just did, you find it confusing, one other way to think about it, I mean you could've ruled out something like this 'cause when you look at this, you could see that okay, her tank holds 32, her tank holds 15 gallons, is you could have tried this one and you would have seen that the units didn't work out. If you have 15 gallons, 15 gallons minus m miles divided by 32 gallons per mile, 32 gallons per mile, what will the units end up being here? Especially right over here, you would have miles divided by gallons per mile, this is going to be minus m times, I guess let me write it this way, m over 32 miles times miles per gallon. Miles per gallon. You get miles squared per gallon, you get all these, you get these crazy units up here. So hopefully it makes sense that we would pick this last choice, and this one almost got me when they said tank of gas. I was thinking, my brain was thinking gallon, I was so already programmed for the answer. But anyway, yes, this is the last, the last choice is what we want to go with.