Given the points \( A(-3,5), B(6,8) \) and \( C(p, q) \), find the values of \( p \) and \( q \) such that \( 2 \overrightarrow{A B}-\overrightarrow{B C}=8 \underline{i}+3 \underline{j} \).

To find the values of \( p \) and \( q \), we start by calculating the vectors involved in the equation \( 2 \overrightarrow{A B} - \overrightarrow{B C} = 8 \underline{i} + 3 \underline{j} \). First, we find \( \overrightarrow{A B} \): \[ \overrightarrow{A B} = \overrightarrow{B} - \overrightarrow{A} = (6 - (-3), 8 - 5) = (6 + 3, 8 - 5) = (9, 3) \] Next, we need to find \( \overrightarrow{B C} \): \[ \overrightarrow{B C} = \overrightarrow{C} - \overrightarrow{B} = (p - 6, q - 8) \] Now we can express \( 2 \overrightarrow{A B} \): \[ 2 \overrightarrow{A B} = 2(9, 3) = (18, 6) \] Next, substitute these values into the equation: \[ (18, 6) - (p - 6, q - 8) = (8, 3) \] Now distribute and set up the equations: \[ (18 - (p - 6), 6 - (q - 8)) = (8, 3) \] This simplifies to: \[ (24 - p, 14 - q) = (8, 3) \] Now we equate the components: 1. \( 24 - p = 8 \) 2. \( 14 - q = 3 \) Solving the first equation for \( p \): \[ p = 24 - 8 = 16 \] Now solving the second equation for \( q \): \[ q = 14 - 3 = 11 \] Thus, the values of \( p \) and \( q \) are: \[ \boxed{(16, 11)} \]