Linear & nonlinear functions: missing value

Find the missing value to make the table represent a linear equation. So let's see this table right over here. So when x is equal to 1, y is 3/2. When x is 2, y is equal to 3. So let's see what happened. When x increased by 1, what did y do? Well looks like y increased by 3 and 1/2 is the same thing as 1 and 1/2. So to go from 1 and 1/2 to 3, it increased by 1 and 1/2, or you could say it increased by 3/2. You could say that 3 is the same thing as 6/2, 6/2 minus 3/2 is another 3/2. All right. Now when we go from 2 to 3, we're increasing by 1 again in x. And what are we doing in y? So we're going from 3, which is the same thing as 6/2 to 9/2. So once again, we are increasing by 3/2. So in order for this to be a linear equation or a linear relationship, every time we increase by 1 in the x direction, we need to increase by 3/2. If we increase by 2, we need to increase by 2 times 3/2. So what are we doing over here on this fourth term on the table? Well we're increasing. We're going from 3 to 8 so we are increasing by 5. So if we're increasing x by 5, then we need to increase y by 5 times 3/2 or 15 over 2. So that's the amount that we have to increase y by. If we started at 9/2 and we're going to increase by 15/2, so it's going to be 9/2 plus 15/2, this is how much we increment by. Remember, we increment 3/2 every time x moves 1. This time, x moved 5. So we're incrementing by 15/2, or you could say we're incrementing by 3/2 five times. But this is going to be equal to 9 plus 15 is 24 over 2 which is equal to 12. And so in the box, we could write 12, and we are done.