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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}

Learn more about this topic:

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Direct Variation: Definition, Formula & Examples

from ACT Prep: Help and Review

Chapter 13 / Lesson 7
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