A= {3a,-4 matriks 5, -2} B={9,7 matriks -7, b} dan C= { 3,-1 matriks -1,2} , Jika A+B = 2c^t, maka nilai a+b =

Given the sets \( A = \{3a, -4, 5, -2\} \), \( B = \{9, 7, -7, b\} \), and \( C = \{3, -1, -1, 2\} \), we are asked to find the value of \( a + b \) under the condition that \( A + B = 2C^T \), where \( C^T \) denotes the transpose of \( C \).

### Step 1: Express \( A + B \)
\[ A + B = \{3a + 9, -4 + 7, 5 - 7, -2 + b\} \]
\[ A + B = \{3a + 9, 3, -2, b - 2\} \]

### Step 2: Compute \( 2C^T \)
The transpose of \( C \), denoted \( C^T \), is the same as \( C \) since it's a set of elements (not a matrix, assuming \( C \) is represented as a vector).
\[ 2C^T = 2 \times \{3, -1, -1, 2\} \]
\[ 2C^T = \{6, -2, -2, 4\} \]

### Step 3: Set the equations \( A + B = 2C^T \)
Equating the corresponding elements:
\[
3a + 9 = 6 \quad \text{(1)}
\]
\[
3 = -2 \quad \text{(2)}
\]
\[
-2 = -2 \quad \text{(3)}
\]
\[
b - 2 = 4 \quad \text{(4)}
\]

### Step 4: Solve the equations
- From equation (1):
  \[
  3a + 9 = 6 \implies 3a = 6 - 9 \implies 3a = -3 \implies a = -1
  \]
- From equation (4):
  \[
  b - 2 = 4 \implies b = 4 + 2 \implies b = 6
  \]

### Step 5: Calculate \( a + b \)
\[
a + b = -1 + 6 = 5
\]

### Final Answer:
\[
\boxed{5}
\]

Would you like more details on any part of this solution? Here are some related questions:

1. What are the properties of vector addition and scalar multiplication?
2. How do you determine the transpose of a matrix or vector?
3. How do operations on sets differ from operations on matrices?
4. What is the importance of element-wise operations in vector mathematics?
5. Can you demonstrate how to solve similar problems with different sets?

**Tip:** Always check the context to confirm whether a vector, matrix, or set operation is being used when interpreting mathematical problems.

Explore the solution to a mathematical problem involving sets A, B, and vector C. Learn how to solve for variables a and b using set operations and vector equations. Suitable for advanced high school students and those interested in algebraic problem-solving.

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