Suppose that \( y \) varies inversely with \( x \), and \( y=3 \) when \( x=4 \). (a) Write an inverse variation equation that relates \( x \) and \( y \). Equation: (b) Find \( y \) when \( x=-15 \)

To find the inverse variation equation, we start with the general form of the equation, which is \( y = \frac{k}{x} \), where \( k \) is a constant. Given that \( y = 3 \) when \( x = 4 \), we can substitute these values to find \( k \): \[ 3 = \frac{k}{4} \] Multiplying both sides by 4 gives: \[ k = 12 \] Thus, the inverse variation equation is: \[ y = \frac{12}{x} \] Now, to find \( y \) when \( x = -15 \): \[ y = \frac{12}{-15} = -\frac{4}{5} \] So, the answers are: (a) \( y = \frac{12}{x} \) (b) \( y = -\frac{4}{5} \) --- Did you know that inverse variation often pops up in real-life situations, like in physics with the relationship between pressure and volume (Boyle’s Law)? As one increases, the other decreases, keeping the product constant. It’s like a seesaw at the playground—when one side goes up, the other side comes down! For those eager to dive deeper into the fascinating world of inverse variations, consider exploring algebra textbooks that feature sections on direct and inverse relationships, or online courses that explain these concepts with interactive examples. A good starting point is Khan Academy, where you'll find quizzes and videos to solidify your understanding!