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In this problem, we are tasked with finding the values of \( a \), \( b \), \( c \), and \( d \) in the matrix equation:

\[
\begin{bmatrix} a & b \\ c & d \end{bmatrix} + 5 \begin{bmatrix} 4 \\ 3 \\ -9 \\ x \end{bmatrix}^T = \begin{bmatrix} -9 & -7 \\ 1 & 2 \end{bmatrix}
\]

### Step 1: Understanding the matrix operations
We start by analyzing the second term on the left side of the equation, which is the scalar multiplication and transposition. The term \( \begin{bmatrix} 4 \\ 3 \\ -9 \\ x \end{bmatrix}^T \) means we need to transpose the vector:

\[
\begin{bmatrix} 4 \\ 3 \\ -9 \\ x \end{bmatrix}^T = \begin{bmatrix} 4 & 3 & -9 & x \end{bmatrix}
\]

Then we multiply this by the scalar \( 5 \):

\[
5 \begin{bmatrix} 4 & 3 & -9 & x \end{bmatrix} = \begin{bmatrix} 20 & 15 & -45 & 5x \end{bmatrix}
\]

### Step 2: Setting up the system of equations
Now we add this result to the matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \):

\[
\begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} 20 & 15 \\ -45 & 5x \end{bmatrix} = \begin{bmatrix} -9 & -7 \\ 1 & 2 \end{bmatrix}
\]

This gives us the following system of equations:
1. \( a + 20 = -9 \)
2. \( b + 15 = -7 \)
3. \( c - 45 = 1 \)
4. \( d + 5x = 2 \)

### Step 3: Solving for \( a \), \( b \), \( c \), and \( d \)

- From equation 1: \( a = -9 - 20 = -29 \)
- From equation 2: \( b = -7 - 15 = -22 \)
- From equation 3: \( c = 1 + 45 = 46 \)
- From equation 4: \( d = 2 - 5x \)

To determine \( d \), you need the value of \( x \). Since the value for \( x \) is not given, you would leave \( d = 2 - 5x \) for now.

### Final Answer:
- \( a = -29 \) (already filled correctly)
- \( b = -22 \)
- \( c = 46 \)
- \( d = 2 - 5x \)

Would you like more details or have any questions? 

Here are five related questions to expand your understanding:
1. How does matrix transposition affect the structure of a matrix?
2. What are the steps involved in scalar multiplication of a matrix?
3. How would the solution change if \( x \) were known?
4. Can this matrix operation be applied in 3D transformations?
5. What is the significance of element-wise addition in matrix equations?

**Tip:** Always ensure to simplify scalar operations and matrix transposition before solving for individual elements in matrix equations.

Find the values of a, b, c, and d in the matrix equation: [ [a b], [c d] ] + 5 * [4 3 -9 x]^T = [ [-9 -7], [1 2] ].

Learn how to find the values of a, b, c, and d in a matrix equation involving scalar multiplication and matrix transposition. This solution walks through each step, explaining matrix operations such as addition, scalar multiplication, and transposition.

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