# The variables x and y are directly proportional, and y = 2 when x = 3. What is the value of y...

## Question:

The variables x and y are directly proportional, and y = 2 when x = 3.

What is the value of y when x = 9?

## Direct Variation:

If x varies directly as y then {eq}x=k y {/eq}, where '{eq}k {/eq}' is a proportionality constant. In this case, {eq}x {/eq} increases (decreases) if {eq}y {/eq} increases (decreases).

The variables {eq}x {/eq} and {eq}y {/eq} are directly proportional.

So by the definition of direct variation,

$$y=k x \,\,\,\,\, \rightarrow (1)$$

Substitute {eq}y=2 \text{ and } x=3 {/eq} in (1) to find {eq}k {/eq}:

$$2 = k (3) \\ \text{Dividing both sides by 3}, \\ k = \dfrac{2}{3}$$

Again, substitute this and {eq}x=9 {/eq} in (1) to find {eq}y {/eq}:

$$y = \dfrac{2}{3} (9) = \boxed{\mathbf{6}}$$