Question: Which equation shows direct variation? Choose 1 answer: A) \( m \cdot n = \frac{1}{3} \) B)…

Which equation shows direct variation?

Choose 1 answer:

A) \( m \cdot n = \frac{1}{3} \)

B) \( m + n = \frac{1}{3} \)

C) \( m \cdot n = 3 \)

D) \( \frac{m}{n} = 3 \)

E) \( 3 \cdot m = \frac{1}{n} \)

Solution

To determine which equation represents direct variation, let’s analyze each option step by step. Direct variation implies that one variable is a constant multiple of the other. Mathematically, this can be expressed as: \[ y = kx \] where \( k \) is the constant of variation. Let’s examine each option: Option A: \[ m \cdot n = \frac{1}{3} \] This equation suggests that the product of \( m \) and \( n \) is constant, which indicates an inverse relationship, not direct variation. Option B: \[ m + n = \frac{1}{3} \] This equation implies that the sum of \( m \) and \( n \) is constant, which does not represent direct variation. Option C: \[ m \cdot n = 3 \] Similar to Option A, this equation shows that the product of \( m \) and \( n \) is constant, indicating an inverse relationship. Option D: \[ \frac{m}{n} = 3 \] This can be rewritten as: \[ m = 3n \] This equation shows that \( m \) is a constant multiple of \( n \), which aligns with the definition of direct variation. Option E: \[ 3 \cdot m = \frac{1}{n} \] This equation can be rearranged to: \[ m = \frac{1}{3n} \] This represents an inverse variation between \( m \) and \( n \). Based on the analysis, Option D is the correct equation that shows direct variation. Answer: D) \(\frac{m}{n} = 3\)