If x varies directly as y, and x = 21 when y = 7, find x when y = 9.

Question:

If x varies directly as y, and x = 21 when y = 7, find x when y = 9.

Direct Variation:

Direct variation shows a relationship between two variables that vary in the same direction. For example, if {eq}y {/eq} varies directly as another variable {eq}z {/eq}, then the equation of variation between the two variables is {eq}y = kz {/eq}.

Answer and Explanation:


If {eq}x {/eq} varies directly as {eq}y {/eq}, we can write this as:

  • {eq}x\propto y {/eq}

Adding a proportionality constant to remove the proportionality sign:

  • {eq}x = ky {/eq}

If {eq}x = 21 {/eq} when {eq}y = 7 {/eq}, then:

  • {eq}21 = k\times 7 {/eq}

Solving for k:

  • {eq}k = \dfrac{21}{7} = 3 {/eq}

Therefore, the equation of variation between the two variables is:

  • {eq}x = 3y {/eq}

Using this equation, the value of {eq}x {/eq} when {eq}y = 9 {/eq} is:

  • {eq}x = 3\times 9 {/eq}
  • {eq}\boxed{\color{blue}{x = 27}} {/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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