From the image you've uploaded, there are two separate matrix equations given. Let's solve each one step-by-step:

Problem 6

Given matrices $A$ and $B$ as follows: $A = \begin{bmatrix} 5 & 7 \\ 2 & 9 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 5 & x \\ 2 & y \end{bmatrix}$ It is stated that $2A = B$.

To find this relation: $2A = 2 \begin{bmatrix} 5 & 7 \\ 2 & 9 \end{bmatrix} = \begin{bmatrix} 10 & 14 \\ 4 & 18 \end{bmatrix}$ Comparing this with $B$, we set: $\begin{bmatrix} 10 & 14 \\ 4 & 18 \end{bmatrix} = \begin{bmatrix} 5 & x \\ 2 & y \end{bmatrix}$ From here, we solve:

• For $x$, $14 = x$.
• For $y$, $18 = y$.

Additionally, calculate:

• $2x + y$
• $2x + 4y$

Calculations:

• $2x + y = 2(14) + 18 = 28 + 18 = 46$
• $2x + 4y = 2(14) + 4(18) = 28 + 72 = 100$

Problem 7

Given matrices $M$ and $N$ as: $M = \begin{bmatrix} q-4 & -8 & 2d-3 \\ b & 2c & -13 \end{bmatrix} \quad \text{and} \quad N = \begin{bmatrix} 3a & -8 & -11 \\ 2b-5 & 3c+8 & -13 \end{bmatrix}$ It's given $M = N$.

From this equality:

1. $q-4 = 3a$
2. $2d-3 = -11$ implies $2d = -8$ so $d = -4$.
3. $b = 2b - 5$ implies $b = 5$.
4. $2c = 3c + 8$ implies $c = -8$.

Finally, find $2a + 2b + 9c$:

• $2a = \frac{2}{3}(q-4)$
• $2b = 2 \times 5 = 10$
• $9c = 9 \times (-8) = -72$

Calculation:

• $2a + 2b + 9c = \frac{2}{3}(q-4) + 10 - 72$

Would you like further details or have any questions on these problems?

Here are some additional questions and a