The matrix equation shown in the image is:

$\begin{pmatrix} a & d \\ b & c \end{pmatrix} = \begin{pmatrix} k+3 & 2k+2 \\ 4k-7 & 3k+3 \end{pmatrix}$ with the condition $a + c = 2b$.

We need to find the value of $d$.

Step 1: Identify corresponding elements

From the matrix equality, we can equate corresponding elements:

$a = k + 3$ $d = 2k + 2$ $b = 4k - 7$ $c = 3k + 3$

Step 2: Use the condition $a + c = 2b$

The condition $a + c = 2b$ gives us the equation:

$(k + 3) + (3k + 3) = 2(4k - 7)$

Simplify both sides:

$4k + 6 = 8k - 14$

Step 3: Solve for $k$

Now solve for $k$:

$4k + 6 = 8k - 14$ $6 + 14 = 8k - 4k$ $20 = 4k$ $k = 5$

Step 4: Find $d$

Now substitute $k = 5$ into the equation for $d$:

$d = 2k + 2 = 2(5) + 2 = 10 + 2 = 12$

So, the value of $d$ is $\boxed{12}$.

Let me know if you need more details or have any questions! Here are some follow-up questions to help expand your understanding:

1. How do you solve matrix equations like this one in general?
2. What are the common methods to solve systems of equations involving conditions?
3. How would this problem change if the condition was $a + c = 3b$?
4. How can you verify the solution after finding $k$?
5. What are the practical applications of matrix equations in real-world problems?

Tip: Always check if conditions are met after solving equations to avoid errors in final answers.