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 can use the given intersection point \( (3, -2) \) in the equations of the lines. Given equations: 1. \( ax + by = 18 \) 2. \( ax - by = 6 \) Substitute the values of \( x = 3 \) and \( y = -2 \) into the equations: 1. \( a(3) + b(-2) = 18 \) 2. \( a(3) - b(-2) = 6 \) Simplify the equations: 1. \( 3a - 2b = 18 \) 2. \( 3a + 2b = 6 \) Now, we have a system of linear equations. We can solve this system to find the values of \( a \) and \( b \). Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}3a-2b=18\\3a+2b=6\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}a=\frac{18+2b}{3}\\3a+2b=6\end{array}\right.\) - step2: Substitute the value of \(a:\) \(3\times \frac{18+2b}{3}+2b=6\) - step3: Simplify: \(18+4b=6\) - step4: Move the constant to the right side: \(4b=6-18\) - step5: Subtract the numbers: \(4b=-12\) - step6: Divide both sides: \(\frac{4b}{4}=\frac{-12}{4}\) - step7: Divide the numbers: \(b=-3\) - step8: Substitute the value of \(b:\) \(a=\frac{18+2\left(-3\right)}{3}\) - step9: Calculate: \(a=4\) - step10: Calculate: \(\left\{ \begin{array}{l}a=4\\b=-3\end{array}\right.\) - step11: Check the solution: \(\left\{ \begin{array}{l}a=4\\b=-3\end{array}\right.\) - step12: Rewrite: \(\left(a,b\right) = \left(4,-3\right)\) The values of \( a \) and \( b \) are \( a = 4 \) and \( b = -3 \).