Writing linear equations in two variables given a table | Algebra 1 (TX TEKS) | Khan Academy

We're told a city bike rental service charges customers based on how long they rent the bicycle. The table shows the total cost for renting a bicycle as a function of the number of rental hours. So they say complete the equation to model the hourly rental cost.

So what they really want us to do is have an equation of Y being, we could say a function of X, that can describe what is going on here. So pause this video and have a go at it before we do this together.

All right, so let's just look at the data a little bit and think about, okay, is this a linear relationship or is this something else? So when we increase our rental hours by two, it looks like here, or when we go from one to three, we're increasing by two and our cost is increasing by... let's see, to go from 12 to 30, it's increasing by 18.

See, when we go from 3 to 5, once again that's two more hours and it looks like every two hours we do indeed increase our cost by 18. Let me check again: hours increase by two and cost indeed increases by 18.

So if we are increasing by $18 every two hours, that's the same thing as you have a change in Y of $9 every time you have a change in X of one hour. So it's $9 per hour. Now, change in Y over change in X might look familiar to you; that is the slope of a line.

So this is a linear relationship. It's going to have the form Y is equal to mx + b, where this is the slope and this is the Y intercept. We just figured out the slope; it is $9 per hour.

So we could say Y is equal to 9x plus whatever the Y intercept is. The Y intercept would be the minimum that they're going to charge you before they even bill you based on hours.

To figure out that, we just have to substitute one of these points. We can say, okay, when X is 1, Y is 12. So let's just substitute that: Y is 12 when X is 1, so 9 * 1 + b, or 12 is equal to 9 + B. You could do this in your head or you could subtract nine from both sides, and you get three is equal to B.

So our equation this right over here is 3, so we get Y is equal to 9x + 3. One way to interpret this is even if you just rent a bike, before they even charge you the hourly, they're going to charge you $3 just to do that, and then they're going to charge you $9 per hour after that.

You can double check that; you could say, okay, well if I had to rent this bike for three hours, I'm going to pay that $3 and then I'm going to pay an extra $27 for the hourly amount that I'm using it. 27 + 3 is indeed $30. You could try out any of these other ones.