Question

The variables x and y vary directly. Use the given values to write an equation that relates x and y. $$x=36, y=12$$

Short answer

The direct variation equation that relates x and y given the values x = 36 and y = 12 is \(y = \dfrac{1}{3}x\).

Step-by-step solution

Understand the Concept of Direct Variation

In mathematics, when we say that x and y vary directly, it means that if x increases, y increases at the same rate and vice versa. If x decreases, y also decreases at the same rate. This is expressed using the equation \(y = kx\) where k is the constant of variation.

Insert the Given Values into the Equation

x and y have been provided as 36 and 12 respectively. We can insert these values into our equation which gives us \(12 = k*36\).

Solve for the Constant of Variation k

If we solve \(12 = k*36\) for k, we get \(\dfrac{12}{36} = k\)

Simplify the Equation to Get the Value of k

Upon simplifying the expression for k, we find that \(k = \dfrac{1}{3}\).

Write the Final Direct Variation Equation

Now, substitute \(k = \dfrac{1}{3}\) back into the direct variation formula. The final equation will be \(y = \dfrac{1}{3}x\).