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You have provided a matrix \( A \) and the problem states that it is symmetric. The matrix \( A \) is:

\[
A = \begin{pmatrix}
1 & 3a + b & 3 \\
17 & 3 & 60 \\
2a - b & 10x & 4
\end{pmatrix}
\]

A symmetric matrix means that \( A = A^T \), meaning that the matrix is equal to its transpose. This implies that corresponding elements across the diagonal are equal, i.e., the element in row \( i \), column \( j \) must equal the element in row \( j \), column \( i \).

### Step 1: Set up the symmetry conditions

1. \( A_{12} = A_{21} \): This gives the equation \( 3a + b = 17 \).
2. \( A_{13} = A_{31} \): This gives the equation \( 3 = 2a - b \).
3. \( A_{23} = A_{32} \): This gives the equation \( 60 = 10x \).

### Step 2: Solve the system of equations

#### From \( 60 = 10x \):
\[
x = 6
\]

#### Solve the system \( 3a + b = 17 \) and \( 3 = 2a - b \):

We now have the system:
\[
3a + b = 17 \quad (1)
\]
\[
2a - b = 3 \quad (2)
\]

Add equations (1) and (2) to eliminate \( b \):
\[
(3a + b) + (2a - b) = 17 + 3
\]
\[
5a = 20
\]
\[
a = 4
\]

Substitute \( a = 4 \) into equation (1):
\[
3(4) + b = 17
\]
\[
12 + b = 17
\]
\[
b = 5
\]

### Step 3: Find \( a + b - x \)

Now that we have \( a = 4 \), \( b = 5 \), and \( x = 6 \), we can calculate \( a + b - x \):
\[
a + b - x = 4 + 5 - 6 = 3
\]

### Final Answer:
\[
a + b - x = 3
\]

Would you like more details on any of these steps or further clarifications?

Here are 5 related questions to expand on this concept:
1. How can you determine if a matrix is symmetric?
2. What are the properties of a symmetric matrix?
3. Can a non-square matrix be symmetric?
4. How would this problem change if the matrix was not symmetric?
5. How does the determinant of a symmetric matrix differ from a general matrix?

**Tip:** Symmetric matrices always have real eigenvalues, which can be a useful property in various applications, including solving systems of equations.

Given the symmetric matrix A, find the value of a + b - x.

Solve for a + b - x in a symmetric matrix by setting up linear equations based on symmetry conditions. This detailed solution explains the algebra involved in solving systems of equations with matrix properties. Suitable for high school students studying algebra and matrices.

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