We are given the system of equations:

\[
9x + 11y = 13 \quad \text{(1)}
\]
\[
11x + 13y = 17 \quad \text{(2)}
\]

### Step 1: Use the method of elimination.

We want to eliminate one of the variables, so let's multiply both equations by suitable numbers to make the coefficients of \( y \) the same.

Multiply equation (1) by 13 and equation (2) by 11:

\[
13(9x + 11y) = 13(13) \quad \Rightarrow \quad 117x + 143y = 169 \quad \text{(3)}
\]
\[
11(11x + 13y) = 11(17) \quad \Rightarrow \quad 121x + 143y = 187 \quad \text{(4)}
\]

### Step 2: Subtract the equations.

Now subtract equation (3) from equation (4):

\[
(121x + 143y) - (117x + 143y) = 187 - 169
\]
\[
4x = 18
\]
\[
x = \frac{18}{4} = 4.5
\]

### Step 3: Substitute the value of \( x \) into one of the original equations.

Substitute \( x = 4.5 \) into equation (1):

\[
9(4.5) + 11y = 13
\]
\[
40.5 + 11y = 13
\]
\[
11y = 13 - 40.5
\]
\[
11y = -27.5
\]
\[
y = \frac{-27.5}{11} = -2.5
\]

### Final Solution:
\[
x = 4.5, \quad y = -2.5
\]

Would you like further clarification on any step?

Here are some related questions for further exploration:
1. How can we solve this system using substitution?
2. What are the advantages of using elimination vs. substitution?
3. Can we use matrices to solve this system?
4. What would happen if both equations had no solution?
5. How would we solve this if the system was nonlinear?

**Tip:** Always double-check your solutions by plugging the values of \( x \) and \( y \) back into the original equations to verify the results.

Learn how to solve the system of linear equations 9x + 11y = 13 and 11x + 13y = 17 using the elimination method. This step-by-step solution covers key algebraic concepts and methods for solving systems of equations, suitable for students in Grades 9-10.

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