# z varies directly as x and inversely as y. If z = 187 when x = 8 and y = 6 , find z if x = 2 and...

## Question:

z varies directly as x and inversely as y. If z = 187 when x = 8 and y = 6 , find z if x = 2 and y = 9.

## Direct and Inverse Variation:

Inverse variation provides a relationship between two variables that change in the opposite direction when all the other variables are held constant. Direct variation, on the other hand, gives a relationship between variables that change in the same direction, when all the other variables are held constant.

## Answer and Explanation:

Given that {eq}z {/eq} varies direcly as {eq}x {/eq} and inversely as {eq}y {/eq}, we can write this relationship as:

• {eq}z\propto \dfrac{x}{y} {/eq}

Removing the proportionality sign and adding a proportionality constant:

• {eq}z = k\cdot \dfrac{x}{y} {/eq}

If {eq}z = 187 {/eq} when {eq}x = 8 {/eq} and {eq}y = 6 {/eq}, then:

• {eq}187 = k\cdot \dfrac{8}{6} {/eq}

Solving for k:

• {eq}k = \dfrac{187\times 6}{8} = 140.25 {/eq}

Therefore:

• {eq}z = 140.25\cdot \dfrac{x}{y} {/eq}

Using the above equation, the value of {eq}z {/eq} when {eq}x = 2 {/eq} and {eq}y = 9 {/eq} is:

• {eq}z = 140.25\cdot \dfrac{2}{9} {/eq}
• {eq}\boxed{\color{blue}{z = \dfrac{187}{6}}} {/eq}