Suppose that \( y \) varies directly with \( x \), and \( y=15 \) when \( x=24 \). \[ \begin{array}{l}\text { (a) Write a direct variation equation that relates } x \text { and } y \text {. } \\ \text { Equation: } \square \\ \text { (b) Find } y \text { when } x=3 \text {. }\end{array} \]

To find the direct variation equation relating \( x \) and \( y \), we start with the proportionality relationship: \[ y = kx \] where \( k \) is the constant of variation. Given \( y = 15 \) when \( x = 24 \), we can solve for \( k \): \[ 15 = k \cdot 24 \] This gives: \[ k = \frac{15}{24} = \frac{5}{8} \] So, the direct variation equation is: \[ y = \frac{5}{8}x \] Now, to find \( y \) when \( x = 3 \): \[ y = \frac{5}{8} \cdot 3 = \frac{15}{8} = 1.875 \] Thus, when \( x = 3 \), \( y \) equals \( 1.875 \). --- Here's a fun fact: In ancient times, the concept of direct variation was linked to the art of trade, where the value of goods exchanged would rise and fall based on direct proportional relationships! Also, in real-world applications, direct variation can be observed in financial situations like salary calculations; if your hourly wage increases, your total pay is directly proportional to the number of hours worked. So, if you up your game in hours, your paycheck reflects it!