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The problem requires obtaining the inverse of the matrix

\[
\begin{pmatrix} 1 & 3 \\ 2 & -2 \end{pmatrix}
\]

and using it to solve the system of equations:

\[
x + 3y = 4 \quad \text{and} \quad 2x - 2y = 9
\]

### Step 1: Finding the Inverse of the Matrix

For a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the inverse is given by:

\[
A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

In this case, the matrix is:

\[
\begin{pmatrix} 1 & 3 \\ 2 & -2 \end{pmatrix}
\]

So, we have:
- \( a = 1 \)
- \( b = 3 \)
- \( c = 2 \)
- \( d = -2 \)

The determinant \( \text{det}(A) = ad - bc \) is:

\[
\text{det}(A) = (1)(-2) - (3)(2) = -2 - 6 = -8
\]

Now, the inverse of the matrix is:

\[
A^{-1} = \frac{1}{-8} \begin{pmatrix} -2 & -3 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} \frac{1}{4} & \frac{3}{8} \\ \frac{1}{4} & -\frac{1}{8} \end{pmatrix}
\]

### Step 2: Solving the System of Equations

We can express the system of equations as:

\[
\begin{pmatrix} 1 & 3 \\ 2 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 9 \end{pmatrix}
\]

Now, multiply both sides of the equation by the inverse matrix \( A^{-1} \):

\[
\begin{pmatrix} x \\ y \end{pmatrix} = A^{-1} \begin{pmatrix} 4 \\ 9 \end{pmatrix}
\]

First, compute:

\[
A^{-1} \begin{pmatrix} 4 \\ 9 \end{pmatrix} = \begin{pmatrix} \frac{1}{4} & \frac{3}{8} \\ \frac{1}{4} & -\frac{1}{8} \end{pmatrix} \begin{pmatrix} 4 \\ 9 \end{pmatrix}
\]

This gives:

\[
\begin{pmatrix} \frac{1}{4}(4) + \frac{3}{8}(9) \\ \frac{1}{4}(4) - \frac{1}{8}(9) \end{pmatrix} = \begin{pmatrix} 1 + \frac{27}{8} \\ 1 - \frac{9}{8} \end{pmatrix} = \begin{pmatrix} \frac{35}{8} \\ \frac{-1}{8} \end{pmatrix}
\]

Thus, the solution is:

\[
x = \frac{35}{8}, \quad y = -\frac{1}{8}
\]

### Final Answer:
- The inverse of the matrix is \( \begin{pmatrix} \frac{1}{4} & \frac{3}{8} \\ \frac{1}{4} & -\frac{1}{8} \end{pmatrix} \).
- The solution to the system of equations is \( x = \frac{35}{8} \) and \( y = -\frac{1}{8} \).

Would you like more details or have any questions?

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Here are 5 related questions:
1. What are the conditions for a 2x2 matrix to have an inverse?
2. How do we calculate the determinant of a 3x3 matrix?
3. Can a matrix with a determinant of zero have an inverse?
4. How can you solve a system of equations using Gaussian elimination?
5. How does Cramer's rule work in solving a system of equations?

**Tip**: The determinant of a matrix gives us key information about whether it is invertible. A determinant of zero means the matrix is singular and has no inverse.

Obtain the inverse of the matrix (1 3, 2 -2) and hence solve the system: x + 3y = 4, 2x - 2y = 9.

Learn how to find the inverse of a 2x2 matrix and use it to solve a system of linear equations. This problem covers key concepts in linear algebra, including matrix inversion and determinants, suitable for students in Grades 10-12.

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