Suppose that \( y \) varies inversely with \( x \), and \( y=3 \) when \( x=6 \). (a) Write an inverse variation equation that relates \( x \) and \( y \). Equation: \( \square \) (b) Find \( y \) when \( x=2 \). \[ y=\square \]

To express the relationship mathematically, when \( y \) varies inversely with \( x \), we can write the equation in the form \( y = \frac{k}{x} \), where \( k \) is a constant. Given that \( y = 3 \) when \( x = 6 \), we can find \( k \) by substituting these values: \[ 3 = \frac{k}{6} \] Multiplying both sides by 6 gives us \( k = 18 \). Thus, the inverse variation equation is: Equation: \( y = \frac{18}{x} \) Now, to find \( y \) when \( x = 2 \), we substitute \( x \) into the equation: \[ y = \frac{18}{2} = 9 \] \[ y = 9 \] The inverse variation equations have fascinated mathematicians for centuries, demonstrating how variables can have a unique relationship where one increases as the other decreases. It was first described in the context of physics when analyzing relationships such as pressure and volume in gases (Boyle's Law), laying the foundation for future scientific innovations! In real-world applications, inverse variation pops up everywhere—from physics to economics! For example, the relationship between the speed of travel and time taken to cover a fixed distance. It’s helpful to understand this concept when planning trips or managing resources, ensuring you make the most out of your time and investments!