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To solve for \( x + y \), let's first interpret the given matrix equation:

\[
\begin{pmatrix} 4 & 2 \\ 5 & 2 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 14 \\ 16 \end{pmatrix}
\]

This matrix multiplication expands to the following system of equations:

1. \( 4x + 2y = 14 \)
2. \( 5x + 2y = 16 \)

We can solve this system by eliminating \( y \). Subtract the first equation from the second:

\[
(5x + 2y) - (4x + 2y) = 16 - 14
\]

Simplifying:

\[
x = 2
\]

Now substitute \( x = 2 \) into the first equation:

\[
4(2) + 2y = 14
\]

\[
8 + 2y = 14
\]

\[
2y = 6
\]

\[
y = 3
\]

Thus, \( x + y = 2 + 3 = 5 \).

So the correct answer is **B) 5**.

Would you like more details on the solution? Here are some related questions:

1. What method can be used to solve systems of equations involving matrices?
2. How do you find the determinant of a 2x2 matrix?
3. What happens if the system has no unique solution?
4. Can this type of matrix equation have an inverse solution method?
5. How does elimination differ from substitution in solving linear systems?

**Tip:** For a matrix equation \( A \mathbf{x} = \mathbf{b} \), you can solve by multiplying both sides by the inverse of \( A \), provided the inverse exists.

Given the following matrix, find the value of x + y.
[ 4 2 ] [ x ] = [ 14 ]
[ 5 2 ] [ y ]   [ 16 ]

Learn how to solve a 2x2 system of linear equations using matrix multiplication. This example covers the key concepts of matrix operations and elimination methods, suitable for students in Grades 9-12.

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