1.) \( \frac{1}{3} x(15 x-12 y)-\frac{1}{2}(6 x-4 y) \) 3. \( \frac{1}{7} p(14 q-21 y)-\frac{1}{5} q(10 p-15 y) \)

Sure, let's simplify both algebraic expressions step by step. --- ### **Problem 1** \[ \frac{1}{3} x (15x - 12y) - \frac{1}{2} (6x - 4y) \] **Step 1: Distribute the terms** 1. Distribute \(\frac{1}{3}x\) into \(15x - 12y\): \[ \frac{1}{3}x \cdot 15x = 5x^2 \] \[ \frac{1}{3}x \cdot (-12y) = -4xy \] 2. Distribute \(-\frac{1}{2}\) into \(6x - 4y\): \[ -\frac{1}{2} \cdot 6x = -3x \] \[ -\frac{1}{2} \cdot (-4y) = 2y \] **Step 2: Combine all terms** \[ 5x^2 - 4xy - 3x + 2y \] **Final Simplified Form:** \[ 5x^2 - 4xy - 3x + 2y \] --- ### **Problem 3** \[ \frac{1}{7} p (14q - 21y) - \frac{1}{5} q (10p - 15y) \] **Step 1: Distribute the terms** 1. Distribute \(\frac{1}{7}p\) into \(14q - 21y\): \[ \frac{1}{7}p \cdot 14q = 2pq \] \[ \frac{1}{7}p \cdot (-21y) = -3py \] 2. Distribute \(-\frac{1}{5}q\) into \(10p - 15y\): \[ -\frac{1}{5}q \cdot 10p = -2pq \] \[ -\frac{1}{5}q \cdot (-15y) = 3qy \] **Step 2: Combine like terms** \[ 2pq - 3py - 2pq + 3qy \] - **Combine \(pq\) terms:** \[ 2pq - 2pq = 0 \] - **Combine \(py\) and \(qy\) terms:** \[ -3py + 3qy = 3qy - 3py \] **Step 3: Factor common terms (optional)** \[ 3y(q - p) \] **Final Simplified Form:** \[ -3py + 3qy \quad \text{or} \quad 3y(q - p) \] --- If you have any further questions or need additional assistance with these problems, feel free to ask!