Linear & nonlinear functions: word problem

Luis and Kate have two video games they want to play. They plan to spend exactly 45 minutes playing the two games. They want to use an equation to express the relationship between the number of minutes they spend playing Super Bologna Man and the number of minutes they spend playing You Have to Cut the Wire. Can this relationship be represented using a linear equation? So let's see if it can. Let's define one variable for the amount of time, the number of minutes they spend playing Super Bologna Man. Let's define that as, well let's just say that's x. So x is equal to time playing Bologna Man. I'll write Bologna right here. And let's define y as y is equal to the number of minutes they spend playing You Have to Cut the Wire. So it's time playing, I'll just call it Wire for short. So if we have the minutes they play time playing Bologna and the time playing Cut Your Wire. If I were to add those two together, so if I say x plus y. I'll write that plus in a neutral color, x plus y. What does this need to be equal to? The time I play Bologna Man plus the time I play Have to Cut the Wire. Well if I add them together, they want to spend exactly 45 minutes playing both games. So this is going to be equal to 45 minutes. Now we have set up an equation that relates the time playing Bologna Man and the time playing You Have to Cut the Wire. But now we have to think about is this a linear relationship? So one way to think about it is the real giveaway for a linear relationship is if you can write it in the traditional form of a line. So if you can write it in the y is equal to mx plus b form, where m is the slope of the line and b is the y-intercept. So let's see if we can do that. Well if we want to do that here, we could just subtract x from both sides. You subtract x from both sides, you get-- so let's subtract an x over here, let's subtract an x over here. So negative x plus this and then subtract an x there. Well, that's going to cancel, and you're going to be left with-- and I'm going try to write it in this form right over here-- y is equal to 45. Let me do that in the same color, just to make it not be confusing. y is equal to, and I'll write the negative x first because we have the x term right over here first. So y is equal to negative x plus 45. All I did is I switched these two terms around. But you see here, it has that form. And you might say wait what is m here? I see that b is 45. Well if I write negative x, that's the same thing as writing negative 1x. So this is definitely a line. I was able to write it in this form right over here. So can this relationship be represented using a linear equation? Absolutely, absolutely yes.