https://file.aiquickdraw.com/mathbot/file/1726259145149xwv7v2my.jpg

Let's break down and solve the equation in the image.

The given matrix equation is:

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

### Step 1: Multiply the two matrices on the left-hand side.
We have:
\[
\left[\begin{array}{cc} 3 & 7 \\ 2 & 4 \end{array}\right] \times \left[\begin{array}{cc} 0 & 2 \\ 5 & 3 \end{array}\right]
\]

To multiply two 2x2 matrices, use the following rule for matrix multiplication:

\[
\left[\begin{array}{cc} a & b \\ c & d \end{array}\right] \times \left[\begin{array}{cc} e & f \\ g & h \end{array}\right] = \left[\begin{array}{cc} ae + bg & af + bh \\ ce + dg & cf + dh \end{array}\right]
\]

Let's compute each element:

- Top-left element: \( 3 \times 0 + 7 \times 5 = 0 + 35 = 35 \)
- Top-right element: \( 3 \times 2 + 7 \times 3 = 6 + 21 = 27 \)
- Bottom-left element: \( 2 \times 0 + 4 \times 5 = 0 + 20 = 20 \)
- Bottom-right element: \( 2 \times 2 + 4 \times 3 = 4 + 12 = 16 \)

Thus, the product of the matrices is:

\[
\left[\begin{array}{cc} 35 & 27 \\ 20 & 16 \end{array}\right]
\]

### Step 2: Set up the equation with 2X.

We now have:

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

Subtract the matrix \(\left[\begin{array}{cc} 35 & 27 \\ 20 & 16 \end{array}\right]\) from both sides:

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

Perform the subtraction element-wise:

\[
2X = \left[\begin{array}{cc} -1 - 35 & -5 - 27 \\ -4 - 20 & 6 - 16 \end

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

This example demonstrates how to solve a 2x2 matrix equation involving matrix multiplication and an unknown matrix X. It covers matrix multiplication rules and basic matrix algebra, suitable for students in Grades 11-12 or undergraduate level.

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