We are given two linear equations:

1. $ax + by = 18$
2. $ax - by = 6$

The point of intersection is given as $(3, -2)$. To find $a$ and $b$, we can substitute $x = 3$ and $y = -2$ into both equations.

Step 1: Substituting into the first equation

$a(3) + b(-2) = 18$ This simplifies to: $3a - 2b = 18 \quad \text{(Equation 1)}$

Step 2: Substituting into the second equation

$a(3) - b(-2) = 6$ This simplifies to: $3a + 2b = 6 \quad \text{(Equation 2)}$

Now, we have a system of equations:

1. $3a - 2b = 18$
2. $3a + 2b = 6$

Step 3: Solving the system of equations

Add the two equations to eliminate $b$: $(3a - 2b) + (3a + 2b) = 18 + 6$ This simplifies to: $6a = 24$ $a = 4$

Step 4: Substituting $a = 4$ into one of the original equations

We can substitute $a = 4$ into either Equation 1 or 2. Let's use Equation 1: $3(4) - 2b = 18$ $12 - 2b = 18$ $-2b = 18 - 12$ $-2b = 6$ $b = -3$

Final Answer:

The values of $a$ and $b$ are: $a = 4, \quad b = -3$

Would you like more details on this solution or have any questions?

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Here are 5 related questions to expand your understanding:

1. How do you solve a system of linear equations with more than two variables?
2. How would the solution change if the point of intersection was different?
3. How can we interpret the slopes of these two lines geometrically?
4. How do we graph the lines $ax + by = 18$ and $ax - by = 6$ after finding $a$ and $b$?
5. What is the method of elimination, and how does it apply to solving systems of equations?

Tip: Always check your solutions by substituting back into the original equations to verify they satisfy both.