Simplify the following expression: \( \frac{3}{2}(x+3 y)-\frac{2}{5}(3 x-y)= \) \( y \) If the number is negative, put the negative sign in the numerator. Answer in

To simplify the expression \( \frac{3}{2}(x + 3y) - \frac{2}{5}(3x - y) \), we start by distributing: 1. Distributing \( \frac{3}{2} \) gives us \( \frac{3}{2}x + \frac{9}{2}y \). 2. Distributing \( -\frac{2}{5} \) yields \( -\frac{6}{5}x + \frac{2}{5}y \). Now we can combine these terms: \[ \left( \frac{3}{2}x - \frac{6}{5}x \right) + \left( \frac{9}{2}y + \frac{2}{5}y \right). \] To combine the \( x \) terms, we need a common denominator, which is 10: \[ \frac{3}{2} = \frac{15}{10} \quad \text{and} \quad -\frac{6}{5} = -\frac{12}{10}, \] so: \[ \frac{15}{10}x - \frac{12}{10}x = \frac{3}{10}x. \] Now, for the \( y \) terms, using 10 as a common denominator gives: \[ \frac{9}{2} = \frac{45}{10} \quad \text{and} \quad \frac{2}{5} = \frac{4}{10}, \] leading to: \[ \frac{45}{10}y + \frac{4}{10}y = \frac{49}{10}y. \] Putting it all together, we get: \[ \frac{3}{10}x + \frac{49}{10}y. \] Now, we set this equal to \(y\): \[ \frac{3}{10}x + \frac{49}{10}y = y. \] To simplify, subtract \( y \) (or \( \frac{10}{10}y \)) from both sides: \[ \frac{3}{10}x + \frac{49}{10}y - \frac{10}{10}y = 0, \] which simplifies to: \[ \frac{3}{10}x + \frac{39}{10}y = 0. \] We can eliminate the fractions by multiplying everything by 10: \[ 3x + 39y = 0. \] Rearranging gives: \[ 3x = -39y \quad \Rightarrow \quad x = -13y. \] This means the final simplified answer is: \[ x + 13y = 0. \]