Direct and inverse variation
Proportionality constant for direct variation
y is directly proportional to x. If y equals 30 when x is equal to 6, find the value of x when y is 45. So let's just take this each statement at a time. y is directly proportional to x. That's literally just saying that y is equal to some constant times x. This statement can literally be translated to y is equal to some constant times x. y is directly proportional to x. Now, they tell us, if y is 30 when x is 6-- and we have this constant of proportionality-- this second statement right over here allows us to solve for this constant. When x is 6, they tell us y is 30 so we can figure out what this constant is. We can divide both sides by 6 and we get this left-hand side is 5-- 30 divided by 6 is 5. 5 is equal to k or k is equal to 5. So the second sentence tells us, this gives us the information that y is not just k times x, it tells us that y is equal to 5 times x. y is 30 when x is 6. And then finally, they say, find the value of x when y is 45. So when y is 45 is equal to-- so we're just putting in 45 for y-- 45 is equal to 5x. Divide both sides by 5 to solve for x. We get 45 over 5 is 9, and 5x divided by 5 is just x. So x is equal to 9 when y is 45.