Suppose that \( y \) varies inversely with \( x \), and \( y=3 \) when \( x=10 \). (a) Write an inverse variation equation that relates \( x \) and \( y \). Equation: (b) Find \( y \) when \( x=5 \) \( y=\square \)

To write the equation, we start with the concept of inverse variation, where we express the relationship as \( y = \frac{k}{x} \) for some constant \( k \). Given that \( y = 3 \) when \( x = 10 \), we can find \( k \) by substituting these values into the equation: \[ 3 = \frac{k}{10} \implies k = 30. \] So the inverse variation equation is: \[ y = \frac{30}{x}. \] Now, to find \( y \) when \( x = 5 \), we substitute \( 5 \) into the equation: \[ y = \frac{30}{5} = 6. \] Thus, \( y = 6 \) when \( x = 5 \). Throughout history, the concept of inverse variation has been applied in various fields, including physics and economics. For instance, the relationship between pressure and volume of a gas held at a constant temperature illustrates this classic inverse variation, known as Boyle's Law. Here, if one quantity increases, the other must decrease to maintain the balance defined by a constant product. To further understand inverse relationships, consider mistakes like misapplying the formula or switching the variables improperly. Always remember that for inverse variation, as one variable increases, the other decreases proportionally. A solid tip is to clearly identify the constant \( k \) whenever provided with specific values to avoid confusion in future problems!