Suppose that \( y \) varies inversely with \( x \), and \( y=10 \) when \( x=2 \). (a) Write an inverse variation equation that relates \( x \) and \( y \). Equation: \( \square \) (b) Find \( y \) when \( x=25 \). \( y=\square \)

To express the relationship between \( y \) and \( x \) in an inverse variation form, we start by using the formula \( y = \frac{k}{x} \), where \( k \) is a constant. Given that \( y=10 \) when \( x=2 \), we can substitute these values to find \( k \): \[ 10 = \frac{k}{2} \implies k = 20 \] The inverse variation equation is thus \[ y = \frac{20}{x} \] Now, to find \( y \) when \( x=25 \), we substitute \( x \) back into the equation: \[ y = \frac{20}{25} = \frac{4}{5} \] So the answers are: Equation: \( y = \frac{20}{x} \) \( y = \frac{4}{5} \) From the perspective of inverse variation, it's fascinating to see how this principle operates in real life! Think about the relationship between speed and time for a given distance: as you double your speed, your travel time halves — they're inversely related! Additionally, it's easy to make errors when substituting values or handling fractions. A common mistake is forgetting to simplify your final answer or misplacing a decimal. Just take a moment to double-check your calculations, and you'll be golden!