Modeling with linear equations: snow

On Monday morning, there were 12 inches of snow on the ground. The weather warmed up, and by Tuesday morning, 2 inches had melted. All right, so we'll have 10 left. 2 more inches melted by Wednesday morning. This pattern continued throughout the week until no more snow was left. So they're essentially saying that we had 12 inches of snow on the ground on Monday and that every day after that, two inches melted. So after Tuesday, you'd have 10 inches, and after Wednesday, you'd have eight inches, and that pattern continued. And what they say is create an equation and a graph to show the relationship between the day and the amount of snow on the ground. So let's define a variable that tells us how far away we are from Monday. So let's let x equal days after Monday. And then let y be equal to inches of snow on the ground. So, one way to think about it is, OK, when x is 0, when we're on Monday, when we're 0 days after Monday, we're going to have 12 inches of snow on the ground, and every day after that, we're going to lose two inches. So if we're on Tuesday, we're going to have 2 inches times 1, because Tuesday is one day, so if x is 1, that means we're on Tuesday. If x is 2, that means we're 2 times 2, we've lost 4 inches, which is what the case is on Wednesday. So this is our equation for the relationship between the day and the amount of snow on the ground. x is the day, how many days after Monday, and then y is the inches of the snow left on the ground. We start with 12, and then every day we lose exactly two inches. Now let's graph this. So I'll make my vertical axis the y-axis, that's inches on the ground. y is equal to inches left on the ground. And then the horizontal axis, that is our x-axis-- let me scroll down a little bit-- this is days after Monday. And so we have 0 days after Monday, we have 1, 2, 3, 4, 5, and 6. And then on Monday, which is exactly 0 days after Monday, that is Monday, we have 12 inches on the ground. So I'll do it up here, so we have 12 inches on the ground right there. And actually, I could do a table if you like. Let me draw this. So if we do x and y, this is the days after Monday, so there's 0, 1, 2, 3, 4, 5, 6. And then on the first day, we have 12 inches, on Monday, 0 days after Monday. Then we lose two inches each day. On day 1 we have 10, day 2, 8, 6, 4, 2, 0. So let's plot these points. We already plotted 0, 12 in that blue color. Now let's plot 1, 10. 1, 10 is right about there. It'll be right over there. Then we can plot 2, 8. So 2 days after Monday, we have 8 inches left on the ground. So this is on Wednesday, so that's 8 inches. And then on 3 days after Monday, we have 6 inches on the ground. You can see that a line is forming here. And then if we go to 4 days after Monday, we have 4 inches on the ground. So that is 4. And then 5 days after Monday, we have 2 inches on the ground. And then finally, on the sixth day, 6 days after Monday-- so what are we at, Sunday now-- we are going to have no inches on the ground. So that's that right there. And you can see that there's this line that formed, because this is a linear relationship. It looks a little curvy because I didn't draw it perfectly, but that is a line. So we've done everything. We've created the equation. We start with 12 inches, every day after that we lose two inches. And we showed a graph that depicts the relationship.