https://file.aiquickdraw.com/mathbot/file/1727293717669r1xvzt46.jpg

The system of equations provided in the image is:

\[
\begin{cases}
2ax + by = 9 \\
bx - 3ay = 4
\end{cases}
\]

It also says that the solution to this system is \( x = 3 \) and \( y = -2 \). The task is to determine the values of \( a \) and \( b \).

### Step 1: Substitute \( x = 3 \) and \( y = -2 \) into both equations.

#### First equation:
\[
2a(3) + b(-2) = 9
\]
This simplifies to:
\[
6a - 2b = 9 \quad \text{(Equation 1)}
\]

#### Second equation:
\[
b(3) - 3a(-2) = 4
\]
This simplifies to:
\[
3b + 6a = 4 \quad \text{(Equation 2)}
\]

### Step 2: Solve the system of equations

We now have two equations:
1. \( 6a - 2b = 9 \)
2. \( 6a + 3b = 4 \)

#### Subtract the first equation from the second:
\[
(6a + 3b) - (6a - 2b) = 4 - 9
\]
This simplifies to:
\[
5b = -5
\]
\[
b = -1
\]

#### Substitute \( b = -1 \) into one of the original equations, for example, the first equation:
\[
6a - 2(-1) = 9
\]
\[
6a + 2 = 9
\]
\[
6a = 7
\]
\[
a = \frac{7}{6}
\]

### Final Answer:
\[
a = \frac{7}{6}, \quad b = -1
\]

Let me know if you'd like further details or clarification!

### Related Questions:
1. How would the solution change if the value of \( x \) or \( y \) were different?
2. What method can you use to solve a system of equations when substitution is not possible?
3. How do you verify that your solution to the system of equations is correct?
4. What role do the coefficients of \( a \) and \( b \) play in determining the solution to the system?
5. How would the problem change if there were more unknowns in the system of equations?

**Tip:** Always double-check your algebraic manipulations when solving systems of equations to avoid small calculation errors.

The system of equations 2ax + by = 9 and bx - 3ay = 4 has the solution x = 3 and y = -2. Determine a and b.

This problem involves solving a system of linear equations with the solution x = 3 and y = -2. Learn how to use substitution and elimination to determine the unknown values of a and b in the equations 2ax + by = 9 and bx - 3ay = 4. Suitable for high school students in Grades 9-12.

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