# Direct variation 1 | Rational expressions | Algebra II | Khan Academy

We are given that the total cost of refueling our car, is directly proportional to how many gallons of gasoline we will buy. Using the first statement, we denote by x – x = the number of gallons purchased, and y will denote – y = the cost of charging the car. In the first statement we are given, that there is a proportional relationship between y and the number of gallons, which are x. This means that y is equal to a constant, we will denote it by k, and we will multiply by x (y = k.x). This is a proportional relationship. If x increases, then y will also increase. We do not know what the value is, but it becomes clear to us. If x decreases, the same will happen with y. Here we are given a little more information, which will help us understand what k is. If a gallon of gasoline costs \$ 2.25, then how many gallons can you buy for \$ 18? If x = 1 – the statement here above for one gallon of gasoline, which says that if we have one gallon and x = 1, then y = 2.25, right? y is the amount for gasoline. We’ve been told that 1 gallon costs \$ 2.25, so I’ll write it down here 2.25 = k.1 I didn’t even have to write multiplied by 1 there. It clearly shows us exactly what is the value, how much is k. We don’t even have to write 1 there. k = 2.25 This is given to us here. So the equation for the relationship between x and y is y = 2.25x, where x is the number of gallons we will buy. y is the value of the purchase, at \$ 2.25 per gallon. Looking for how many gallons we can to buy for 18 dollars? \$ 18 will be the value of the entire purchase. This is y, y is the cost of refueling. So 18 = 2.25x To find x, we divide the two sides by 2.25, let’s do it. Divide 18 by 2.25, divide 2.25x by 2.25, and what do we get? I’ll go a little further down. On the right 2.25 is shortened and remains x. And how much is 18 / 2.25? I will write it here. Let’s look at it as a liver. 2.25 is the same – just to write it here – 2.25 = 2 and 1/4, which is the same as 9/4 In other words 18 / 2.25 = 18 / (9/4), which is equal to 18x (4/9), or 18/1 x 4/9. Let’s see, 18/9 = 2, 9/9 = 1 And after simplification we get 8. 18 / 2.25 = 8, so we can buy 8 gallons for \$ 18. 