Question

The variables x and y vary directly. Use the given values to write an equation that relates x and y. $$x=27, y=3$$

Short answer

The equation that relates x and y is \(y = \frac{1}{9}x\).

Step-by-step solution

Understanding Direct Variation

In direct variation, the two variables are directly proportional to each other. Therefore, we apply the formula for direct variation, which is \(y = kx\), where \(k\) is the constant of proportionality.

Substituting Given Values

Next, we substitute the given values of \(x\) and \(y\) into the formula. We have \(x = 27\) and \(y = 3\), so we substitute these values to get \(3 = k * 27\).

Solving for the constant of variation, \(k\)

To isolate \(k\), we divide both sides of the equation by 27. This gives us \(k = \frac{3}{27}\)

Simplifying the value of \(k\)

Simplifying the fraction \(\frac{3}{27}\), we get \(k = \frac{1}{9}\).

Writing the Equation

Finally we substitute the value of \(k\) we've found into the original equation of a direct variation to get our relation: \(y = \frac{1}{9}x\). This is the equation that relates \(x\) and \(y\).