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We have the matrix equation:

\[
\begin{bmatrix} 
3 & 7 \\
2 & 4
\end{bmatrix}
\begin{bmatrix} 
0 & 2 \\
5 & 3 
\end{bmatrix} + 2X = 
\begin{bmatrix} 
1 & -5 \\
-4 & 6 
\end{bmatrix}
\]

Let's break it down into steps.

### Step 1: Multiply the two matrices on the left-hand side.
Multiply:

\[
\begin{bmatrix} 
3 & 7 \\
2 & 4
\end{bmatrix}
\text{ and }
\begin{bmatrix} 
0 & 2 \\
5 & 3 
\end{bmatrix}
\]

The resulting matrix is obtained by performing the dot product between rows of the first matrix and columns of the second matrix. So:

- First row, first column: \( (3 \times 0) + (7 \times 5) = 0 + 35 = 35 \)
- First row, second column: \( (3 \times 2) + (7 \times 3) = 6 + 21 = 27 \)
- Second row, first column: \( (2 \times 0) + (4 \times 5) = 0 + 20 = 20 \)
- Second row, second column: \( (2 \times 2) + (4 \times 3) = 4 + 12 = 16 \)

Thus, the product is:

\[
\begin{bmatrix} 
35 & 27 \\
20 & 16
\end{bmatrix}
\]

### Step 2: Substitute the product back into the equation.

Now, the equation becomes:

\[
\begin{bmatrix} 
35 & 27 \\
20 & 16
\end{bmatrix} + 2X = 
\begin{bmatrix} 
1 & -5 \\
-4 & 6 
\end{bmatrix}
\]

### Step 3: Isolate \(2X\).
Subtract the matrix \( \begin{bmatrix} 35 & 27 \\ 20 & 16 \end{bmatrix} \) from both sides:

\[
2X = 
\begin{bmatrix} 
1 & -5 \\
-4 & 6 
\end{bmatrix} - 
\begin{bmatrix} 
35 & 27 \\
20 & 16
\end{bmatrix}
\]

Perform element-wise subtraction:

- First row, first column: \( 1 - 35 = -34 \)
- First row, second column: \( -5 - 27 = -32 \)
- Second row, first column: \( -4 - 20 = -24 \)
- Second row, second column: \( 6 - 16 = -10 \)

This gives:

\[
2X = 
\begin{bmatrix} 
-34 & -32 \\
-24 & -10
\end{bmatrix}
\]

### Step 4: Solve for \(X\).
Divide every element of the matrix by 2:

\[
X = \frac{1}{2} \begin{bmatrix} 
-34 & -32 \\
-24 & -10
\end{bmatrix}
\]

Thus, we get:

\[
X = \begin{bmatrix} 
-17 & -16 \\
-12 & -5
\end{bmatrix}
\]

This is the solution for \(X\).

Would you like further details or clarifications? 

Here are some related questions:
1. What are the rules for matrix multiplication?
2. How do you solve systems of linear equations using matrices?
3. What is the role of the identity matrix in solving matrix equations?
4. How does matrix addition differ from matrix multiplication?
5. What are some common applications of matrices in real-world problems?

**Tip:** When multiplying matrices, ensure that the number of columns in the first matrix matches the number of rows in the second matrix for the multiplication to be defined.

Solve the matrix equation: [ [3, 7], [2, 4] ] * [ [0, 2], [5, 3] ] + 2X = [ [1, -5], [-4, 6] ]

Learn how to solve the matrix equation [ [3, 7], [2, 4] ] * [ [0, 2], [5, 3] ] + 2X = [ [1, -5], [-4, 6] ] with a step-by-step guide. Understand matrix multiplication, addition, and solving for an unknown matrix X. Suitable for advanced high school or undergraduate students.

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