15. \( 6 y^{2}=3 y^{2}+48 \)

To solve the equation \( 6y^{2} = 3y^{2} + 48 \), start by simplifying the equation. Subtract \( 3y^{2} \) from both sides: \[ 6y^{2} - 3y^{2} = 48 \] which simplifies to \( 3y^{2} = 48 \). Now, divide both sides by 3: \[ y^{2} = 16 \] Taking the square root of both sides, we find the solutions: \[ y = 4 \quad \text{or} \quad y = -4 \] So the complete solution set is \( y = 4 \) and \( y = -4 \). Want to see it in action? Imagine you’re solving a mystery: each step you take gets you closer to unmasking the solution. Anytime you see terms you can combine, it’s like getting rid of distractions to focus on your prize. It's important to check your work! After solving for \( y \), plug those numbers back into the original equation to ensure they satisfy it. One common mistake is to forget the negative solution; remember, \( y = -4 \) is just as valid as \( y = 4 \).