If y is directly proportional (x+2)(x+5) and y= 6 when x = 1, find y when x = 4. y =

Answer 1

"Directly proportional" means that we can write y as a multiple of the expression with the x's. Since we do not know exactly what multiple it is right now, we'll need to use a variable to show this, and most commonly we use the letter k. Specifically, we have that

y = k(x+2)(x+5)

Right now, we do not know what k is, but that's why they give us some extra info. We know that when x=1 we should have y=6. Let's plug this into the expression we set up above:

6 = k(1+2)(1+5)

6 = k(3)(6)

6 = 18k

Notice that because we plugged in numbers for x and y, the only part left unknown is k, and we can solve for it! To get k by itself, we divide both sides by 18 to find that:

k = 1/3

Going back to the original expression, we now know that

y = 1/3(x+2)(x+5)

This gives us complete information about how x and y are related, so in order to finish the problem, all we have to do is plug in x=4:

y = 1/3(4+2)(4+5)

y = 1/3(6)(9)

y = 2(9)

y = 18 when x = 4.