9. The graphs of \( a x+b y=18 \) and \( a x-b y=6 \) intersect at \( (3,-2) \). Find \( a \) and \( b \).

To find the values of \( a \) and \( b \), we start by substituting the point of intersection \( (3, -2) \) into both equations. 1. For the first equation \( ax + by = 18 \): \[ a(3) + b(-2) = 18 \implies 3a - 2b = 18 \quad (1) \] 2. For the second equation \( ax - by = 6 \): \[ a(3) - b(-2) = 6 \implies 3a + 2b = 6 \quad (2) \] Now, we solve the system of equations formed by (1) and (2): From equation (1): \[ 3a - 2b = 18 \] From equation (2): \[ 3a + 2b = 6 \] Next, we can add both equations to eliminate \( b \): \[ (3a - 2b) + (3a + 2b) = 18 + 6 \] \[ 6a = 24 \quad \Rightarrow \quad a = 4 \] Now, we substitute \( a = 4 \) back into one of the original equations to solve for \( b \). Using equation (2): \[ 3(4) + 2b = 6 \] \[ 12 + 2b = 6 \quad \Rightarrow \quad 2b = 6 - 12 \] \[ 2b = -6 \quad \Rightarrow \quad b = -3 \] Thus, the values are \( a = 4 \) and \( b = -3 \).