Y varies directly as X. If y=18 when x=6, how do you find the formula, then find y when x is 27?

Answer 1

Think back for a sec to elementary word problems, like "Two numbers add up to 78. One is twice the other." The information is given to us in English, and our first step is always just to translate those two English sentences into math "sentences" called equations, using variables to represent anything unknown (in this example, the first sentence translates to a + b = 78, and the second to a = 2b).

In your problem, our task is the same; first, we just need to express all of our given information in the language of math. "Y varies directly as x" means that "y is always ____ times bigger than x." For one problem, it might turn out that y is always exactly 3 times bigger than x, for another y might actually be smaller than x by a factor of 0.2, but usually we don't know upfront what that number (called the proportionality constant if you wanna be fancy) is. So in general we just say "y is always k times x," using that k as a placeholder for whatever number we scale up x by to get y. That means whenever you see the English words "y varies directly as x," you should translate that into the equation:

y = kx y is always k times x

I know it may seem like the last thing we want to do when we already have two variables is to add a third, but trust me, this is by far the easiest and most organized way to solve these problems.

Okay, so we've got the first English sentence of our problem written mathematically. We can add the information from the second sentence of the problem, "y=18 when x=6," by plugging in those two values:

y = k*x

18 = k*6

Now all of a sudden our equation with three variables just has one! See if you can solve that equation for k from there; all you have to do is divide and simplify.

Once you have your value for k, the proportionality constant, remember that for the purposes of this problem y is always k times x, so you can go back to our original formula and permanently replace the letter k we were using as a placeholder with your value. So, for example, if I'd solved for k and found that k = -4 (it doesn't for your problem), I would go back and update my formula like this:

Old, not super helpful formula: y = kx

New, super amazing formula: y = -4x

Now I can plug in any x value I want to my new and improved formula and it will spit out the corresponding y value. For example, using -4 as our constant of variation, I would find y when x is 27 by just plugging in 27 for x like this:

y = -4x

y = -4(27)

y = -108

See if you can do the same with the value you found for k! And feel free to respond here if you want to check your answer or have any further questions.

Additional resources and videos that might be helpful:

https://www.khanacademy.org/math/algebra-home/alg-rational-expr-eq-func/alg-direct-and-inverse-variation/v/proportionality-constant-for-direct-variation

https://www.freemathhelp.com/direct-variation.html

https://youtu.be/cFAE8RiKtj8

Answer 2

I always looked at these problems as 3 steps

1) Write the general form of the equation. If y varies directly with x, y = kx, where k is the constant of variation.

2) Use the x and y values given to find the value of k (y=18 when x=3)

18 = k(3). If you divide both sides by 3, k=6. And re-write the equation using that value of k,

y = 6x

3) Finally, solve the given problem, what is y when x is 27.

y = 6(27)

y = 162