Constructing linear equation from context

Tara was hiking up a mountain. She started her hike at an elevation of 1200 meters and ascended at a constant rate. After four hours, she reached an elevation of 1700 meters. Let y represent Tara's elevation in meters after x hours.

And they ask us, this is from an exercise on Khan Academy, says complete the equation for the relationship between the elevation and the number of hours. And if you're on Khan Academy, you would type it in. But we could do it by hand, so pause this video and work it out on some paper, and let's see if we get to the same place.

All right, now let's do this together. So, first of all, they tell us that she's ascending at a constant rate. So that's a pretty good indication that we could describe her elevation based on the number of hours she travels with a linear equation.

And we can even figure out that constant rate. It says that she goes from 1200 meters to 1700 meters in four hours. So we could say her rate is going to be her change in elevation over a change in time. So her change in elevation is 1700 meters minus 1200 meters, and she does this over four hours.

Her change in time is four hours, so her constant rate in the numerator here, 1700 minus 1200 is 500 meters. She's able to go up 500 meters in four hours. If we divide 500 by 4, this is 125 meters per hour.

And so we could use this now to think about what our equation would be. Our elevation y would be equal to... Well, where is she starting? Well, she's starting at 1200 meters. So she's starting at 1200 meters, and then to that, we're going to add how much she climbs based on how many hours she's traveled.

So it's going to be this rate, 125 meters per hour, times the number of hours she has been hiking. So the number of hours is x times x. So this right over here is an equation for the relationship between the elevation and the number of hours.

Another way you could have thought about it, you could have said, okay, this is going to be a linear equation because she's ascending at a constant rate. You could say the slope-intercept form for a linear equation is y is equal to mx plus b, where b is your y-intercept. What is the value of y when x is equal to zero?

And you say, all right, when x is equal to zero, she's at an elevation of 1200. Then m is our slope, so that's the rate at which our elevation is increasing. And that's what we calculated right over here; our slope is 125 meters per hour.

So notice these are equivalent. I've just... these two terms are swapped. So we could either write y is equal to 1200 plus 125x, or you could write it the other way around. You could write 125x plus 1200. They are equivalent.