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The given matrix equation is:

\[
\begin{bmatrix}
\frac{a+b}{9} & \frac{2-a}{5-c}
\end{bmatrix} =
\begin{bmatrix}
\frac{b+a}{9} & \frac{2a+8}{-c+5}
\end{bmatrix}
\]

### Solving for \( a \):

The matrices are equal, so the corresponding elements must be equal. This gives two equations:

1. \(\frac{a+b}{9} = \frac{b+a}{9}\)
2. \(\frac{2-a}{5-c} = \frac{2a+8}{-c+5}\)

#### Step 1: Solve the first equation
\[
\frac{a+b}{9} = \frac{b+a}{9}
\]

This equation is always true because the left-hand side and the right-hand side are identical. Thus, this equation gives no additional information.

#### Step 2: Solve the second equation
\[
\frac{2-a}{5-c} = \frac{2a+8}{-c+5}
\]

Cross-multiply:
\[
(2-a)(-c+5) = (2a+8)(5-c)
\]

Expand both sides:
\[
-2c + 10 + ac - 5a = 10a + 40 - 2ac - 8c
\]

Simplify:
\[
-2c + 10 + ac - 5a = 10a + 40 - 2ac - 8c
\]

Combine like terms:
\[
ac + 2ac - 5a - 10a = -8c + 2c + 40 - 10
\]

\[
3ac - 15a = -6c + 30
\]

Factor:
\[
3a(c - 5) = -6(c - 5)
\]

Divide through by \((c-5)\), assuming \(c \neq 5\):
\[
3a = -6
\]

Solve for \(a\):
\[
a = -2
\]

### Final Answer:
\[
a = -2
\]

Let me know if you need further clarification or additional steps! Here are some related questions to explore:

1. What is the value of \( b \) in this equation?
2. How does the solution change if \( c = 5 \)?
3. How can we verify the solution by substituting back into the original equation?
4. Are there any constraints on \( c \) or \( b \)?
5. What methods can be used to solve similar matrix equations systematically?

**Tip:** Always check the solution by substituting it back into the original equation to confirm its correctness.

Solve the matrix equation: \[
\begin{bmatrix}
\frac{a+b}{9} & \frac{2-a}{5-c}\end{bmatrix} = \begin{bmatrix}
\frac{b+a}{9} & \frac{2a+8}{-c+5}\end{bmatrix}\]

Learn how to solve the matrix equation \( \begin{bmatrix} \frac{a+b}{9} & \frac{2-a}{5-c} \end{bmatrix} = \begin{bmatrix} \frac{b+a}{9} & \frac{2a+8}{-c+5} \end{bmatrix} \) and find the value of \( a \). This solution involves matrix equality, cross-multiplication, and simplifying rational expressions. Suitable for high school students in Grades 9-12.

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