# Suppose that y varies directly with x and inversely with z. If y= 25 when x= 35 and z= 7, write...

## Question:

Suppose that y varies directly with x and inversely with z. If y= 25 when x= 35 and z= 7, write the equation that models the relationship. Then find y when x= 12 and z= 4.

## Direct and Inverse Variation:

Direct variation represents two variables that change in the same direction. That is, variables that are related in such a way that an increase/decrease in one variable causes an increase/decrease in the other variable. Inverse variation, on the other hand, represents a relationship between two variables which vary in the opposite direction.

If {eq}y {/eq} varies direcly as {eq}x {/eq} and inversely as {eq}z {/eq}, we can write this relation as:

• {eq}y\propto \dfrac{x}{z} {/eq}.

Removing the proportionality sign and adding a proportionality constant, we have:

• {eq}y = \dfrac{kx}{z} {/eq}

Given that {eq}y = 25 {/eq} when {eq}x = 35 {/eq} and {eq}z = 7 {/eq}, the proportionality constant will be equal to:

• {eq}25 = \dfrac{35k}{7} {/eq}
• {eq}25\times 7 = 35k {/eq}
• {eq}k = \dfrac{25\times 7}{35} = 5 {/eq}

Thus, an equation representing the relationship between the 3 variables is:

• {eq}y = \dfrac{5x}{z} {/eq}

Using the above relation, the value of y when x = 12 and z = 4 is equal to:

• {eq}y = \dfrac{5\times 12}{4} = \boxed{15} {/eq} 