# y varies directly as x, and y = 20 when x = 5. Write a function that models the variation.

## Question:

{eq}y {/eq} varies directly as {eq}x {/eq}, and {eq}y = 20 {/eq} when {eq}x = 5 {/eq}.

Write a function that models the variation.

## Direct Variation:

Direct variation explains the relationship between two variables that change in the same direction. That is, when one variable increases, the other variable increase as well. For example, the pressure and temperature of a gas. Increasing the temperature of a gas in a container increases the gas volume and reducing the pressure reduces the volume.

Given that the variable {eq}y {/eq} varies directly as the variable {eq}x {/eq}, we can write this as:

• {eq}y\propto x {/eq}

To remove the proportionality sign, we will put an equal sign and add a constant of variation.

• {eq}y = kx {/eq}

We are given that when {eq}y = 20 {/eq}, {eq}x = 5 {/eq}. Therefore, the constant of variation is equal to:

• {eq}20 = k\times 5 {/eq}
• {eq}k = \dfrac{20}{5} = 4 {/eq}

Therefore, the function that models the relationship between {eq}\rm x\; and\; y {/eq} is equal to:

• {eq}\boxed{y = 4x} {/eq} 