Constructing linear equations from context

tarah was hiking up a mountain she started her hike at an elevation of 1,200 meters and ascended at a constant rate after four hours she reached an elevation of 1,700 meters let Y represent Terra's elevation in meters after X hours and they asked us this is from a 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 out 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 are we could describe her elevation based on the number of hours she travels with a linear equation and we could even figure out that constant rate it says that she goes from 1200 meters to 1700 meters in 4 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 4 hours over her change in time is 4 hours so her constant rate in the numerator here 1700 minus 1200 is 500 meters she's able to go up 500 meters in 4 hours if we divide 500 divided 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 it'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 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 0 and you say all right when X is equal to 0 she's at an elevation of 1,200 and 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 have these two terms or swapped so we could either write Y is equal to 1200 plus 125 X or you could write it the other way around you could write 125 X plus 1200 they are equivalent