If y varies directly as x, find the constant of variation k and the direct variation equation for the situation. y=3 when x=6
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Direct variation means that when one variable gets larger/smaller, the other variable gets proportionally larger/smaller.
So all direct variation problem can use the form y = kx, where k is a number that is consistent for the entire problem.
Such as y = 2x, this means that y is twice as large as x. If x gets larger by 4 y gets larger by 8. If x gets larger by 10, y gets larger by 20....and so on.
We have to find the k for your problem, and it gives us a big clue. It says x = 6 and y =3 Therefore y is 1/2 the size of x. So our constant of variation, k, is 1/2.
Mathematically we could have found the 1/2 by plugging in the y and x values and solving for k.
3 = k(6). Solve for k by dividing by 6 to each side and k = 3/6 = 1/2 *This is important to know how to do since k is not always as nice as 1/2. What if it was 0.3542 or something? You wouldn't know it was unless you solved the equation way.
The direct variation equation is y = (1/2) x. This is just y = kx with the k plugged in. Depending on the problem your "k" value will change.