Verified Answer :

  • Identify direct variation as the relationship \(y = kx\) where \(k\) is a constant.
  • Analyze each option: \(y = \frac{x}{9}\), \(y = 0.5x\), and \(y = 2x\) can all be written in the form \(y = kx\).
  • Recognize \(y = \frac{9}{x}\) as an inverse variation, meaning it cannot be written in the form \(y = kx\).
  • Conclude that the equation failing to show direct variation is \(\boxed{y = \frac{9}{x}}\).

Explanation

  1. Understanding Direct Variation

Direct variation exists between two variables \(x\) and \(y\) if there is a non-zero constant \(k\) such that the relationship between them can be written as \(y = kx\). This equation indicates that as \(x\) increases or decreases, \(y\) changes in direct proportion to \(x\). If an equation cannot be expressed in this form, it does not represent a direct variation. We examine the four given equations to check if they fit this definition:

  1. \(y = \frac{9}{x}\)
  2. \(y = \frac{x}{9}\)
  3. \(y = 0.5x\)
  4. \(y = 2x\)
  1. Evaluating Each Equation

We compare each equation to the standard form \(y = kx\):

  • For \(y = \frac{x}{9}\), we can rewrite it as \(y = \frac{1}{9} \cdot x\), which matches the form \(y = kx\) with \(k = \frac{1}{9}\). This is a direct variation.
  • For \(y = 0.5x\), this already matches the form \(y = kx\) with \(k = 0.5\). This is a direct variation.
  • For \(y = 2x\), this matches the form \(y = kx\) with \(k = 2\). This is a direct variation.
  • For \(y = \frac{9}{x}\), this equation relates \(y\) and \(x\) by division, which is characteristic of inverse variation (specifically, \(y = \frac{k}{x}\)). It cannot be written as \(y = kx\) for any constant \(k\). Thus, this equation does not represent a direct variation.
  1. Conclusion

Based on the evaluation, the equation \(y = \frac{9}{x}\) is the only one that does not follow the form \(y = kx\). Therefore, it does not represent a direct variation between \(x\) and \(y\). The final answer is \(y = \frac{9}{x}\).

Examples

Direct variation is commonly used in daily life to describe relationships where two quantities scale together. For example, if you are buying apples at a store and they cost \$\(2\) per apple, the total cost \(C\) is given by \(C = 2n\), where \(n\) is the number of apples. As you buy more apples, the cost increases proportionally. This is a clear example of direct variation, where the price per apple represents the constant \(k\). Understanding these variations helps in budgeting, calculating fuel consumption relative to distance, or even understanding how speed affects travel time when other factors remain constant.

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