\( y \) varies directly with \( x \). \( y \) is 30 when \( x \) is 6 . What is the value of \( k \) ? \( y=[?] x \)

To find the value of \( k \), we use the direct variation formula, which is \( y = kx \). Given that \( y \) is 30 when \( x \) is 6, we can substitute these values into the equation: \[ 30 = k \cdot 6 \] To solve for \( k \), divide both sides by 6: \[ k = \frac{30}{6} = 5 \] So the equation is \( y = 5x \). In the world around us, direct variation pops up in various contexts! For instance, if you’re cooking and want to scale up a recipe, you’re directly varying the amount of each ingredient according to a constant ratio. If you double the number of servings, you simply double each ingredient to maintain that tasty balance—just like how \( y \) varies directly with \( x \)! When tackling problems involving direct variation, a common mistake is forgetting to find the constant of variation \( k \) before jumping into other calculations. Always make sure you've established this crucial relationship first. Additionally, ensure you're solving for \( y \) or \( x \) correctly—watch out for mix-ups that can lead to incorrect interpretations of your variables!