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We are given two matrices, \( A \) and \( B \), and we are told that they are equal. The problem asks us to find the value of \( a + b + c + d \).

The matrices are as follows:

\[
A = \begin{pmatrix} 
7 & b + 3 \\ 
d - 5 & -b - 3 
\end{pmatrix}
\]
\[
B = \begin{pmatrix} 
7c & 3a - 1 \\ 
2a + 1 & -5c 
\end{pmatrix}
\]

Since \( A = B \), the corresponding elements must be equal. Let's equate the elements in the matrices:

1. Top-left element: 
   \[
   7 = 7c \quad \Rightarrow \quad c = 1
   \]

2. Top-right element: 
   \[
   b + 3 = 3a - 1 \quad \Rightarrow \quad b = 3a - 4
   \]

3. Bottom-left element: 
   \[
   d - 5 = 2a + 1 \quad \Rightarrow \quad d = 2a + 6
   \]

4. Bottom-right element: 
   \[
   -b - 3 = -5c \quad \Rightarrow \quad -b - 3 = -5 \quad \Rightarrow \quad b = 2
   \]

Now that we know \( b = 2 \), substitute this into the equation \( b = 3a - 4 \) to solve for \( a \):

\[
2 = 3a - 4 \quad \Rightarrow \quad 3a = 6 \quad \Rightarrow \quad a = 2
   \]

Now substitute \( a = 2 \) into the equation \( d = 2a + 6 \) to find \( d \):

\[
d = 2(2) + 6 = 4 + 6 = 10
   \]

We now have the values:
- \( a = 2 \)
- \( b = 2 \)
- \( c = 1 \)
- \( d = 10 \)

Finally, calculate \( a + b + c + d \):

\[
a + b + c + d = 2 + 2 + 1 + 10 = 15
\]

Thus, the value of \( a + b + c + d \) is **15**.

Would you like further details or explanations? 

Here are five related questions:
1. How do we determine if two matrices are equal?
2. Can matrices with different dimensions be equal?
3. What happens if the system of equations for matrix equality has no solution?
4. How do you interpret negative values in matrix elements?
5. Can matrix elements represent real-world quantities?

**Tip:** When solving systems of equations from matrices, always verify that each equation is consistent before substituting values.

Diketahui matriks A dan matriks B sebagai berikut: A = [[7, b+3], [d-5, -b-3]] dan B = [[7c, 3a-1], [2a+1, -5c]]. Jika matriks A sama dengan B maka tentukan nilai a + b + c + d.

Solve the matrix equality problem where matrices A and B are equal. This involves solving simultaneous equations to find the values of a, b, c, and d. Suitable for high school students studying algebra and matrices.

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