We are tasked with solving for $a$ and $b$ based on the system of equations where the solution point is $(2, -3)$.

The equations are:

1. $ax + by + c = 0$
2. $bx - ay - 12 = 0$

Substitute $x = 2$ and $y = -3$ into both equations:

Step 1: Substituting into the first equation

$a(2) + b(-3) + c = 0$ 2a - 3b + c = 0 \tag{1}

Step 2: Substituting into the second equation

$b(2) - a(-3) - 12 = 0$ 2b + 3a - 12 = 0 \tag{2}

Step 3: Solving the system of equations

We now solve for $a$, $b$, and $c$ from the two equations:

From Equation (1): c = -2a + 3b \tag{3}

Substitute $c = -2a + 3b$ into Equation (2): $2b + 3a - 12 = 0$

Simplify: 3a + 2b = 12 \tag{4}

We now have the system of equations:

1. $2a - 3b + c = 0$
2. $3a + 2b = 12$

Solving these equations will yield the values of $a$ and $b$. Let me compute this for you. sympy import symbols, Eq, solve

Define variables

a, b, c = symbols('a b c')

Define equations

eq1 = Eq(2a - 3b + c, 0) # 2a - 3b + c = 0 eq2 = Eq(3a + 2b, 12) # 3a + 2b = 12

Solve for a, b, c

solution = solve((eq1, eq2), (a, b, c)) solutionThe solutions for $a$ and $b$ in terms of $c$ are:

$a = \frac{36}{13} - \frac{2c}{13}$ $b = \frac{24}{13} + \frac{3c}{13}$

To find the specific values of $a$ and $b$, we would need a particular value for $c$. If more details about $c$ are provided, I can calculate exact values.

Would you like to specify $c$, or should we proceed with additional steps or a general explanation? Here are five follow-up questions:

1. Should we assume $c$ is zero for simplicity?
2. Would you like a deeper explanation of how these equations were solved?
3. Are there additional constraints for this problem, such as $c$?
4. Do you need this solution verified through substitution?
5. Should we plot the solutions or explore alternate approaches?

Tip: Providing constraints on all unknowns can make solving linear systems easier.