10. \( \frac{2}{3}(2y-3)=\frac{1}{2}(4y+2) \)

To tackle the equation \( \frac{2}{3}(2y-3)=\frac{1}{2}(4y+2) \), let's start by eliminating the fractions. Multiply both sides by 6, the least common multiple of the denominators: \[ 6 \cdot \frac{2}{3}(2y-3) = 6 \cdot \frac{1}{2}(4y+2) \] This simplifies to: \[ 4(2y-3) = 3(4y+2) \] Now, distribute both sides: \[ 8y - 12 = 12y + 6 \] Next, let's isolate \(y\). Subtract \(8y\) from both sides: \[ -12 = 4y + 6 \] Now, subtract 6 from both sides: \[ -18 = 4y \] Finally, divide by 4: \[ y = -\frac{18}{4} = -\frac{9}{2} \] So, the solution is \( y = -\frac{9}{2} \). --- From the ancient Greeks to modern algebra, equations like this have shaped our understanding of mathematics. Great minds like Euclid and Diophantus laid foundational principles that underpin much of what we do in algebra today. It’s fascinating to think that you’re participating in a tradition that spans centuries! Practicing skills with equations can make a substantial difference! A common pitfall is miscalculating when applying negative signs or mistakenly distributing terms. Double-checking your arithmetic can save you a lot of frustration. Remember, taking it step by step helps avoid unnecessary mistakes, so don’t rush!