Let's solve the given system of linear equations step by step:
[tex]\[
\begin{array}{l}
x + 2y = 5 \\
2x + y = 16
\end{array}
\][/tex]
Step 1: Solve the first equation for [tex]\( x \)[/tex]:
[tex]\[
x + 2y = 5 \implies x = 5 - 2y
\][/tex]
Step 2: Substitute [tex]\( x = 5 - 2y \)[/tex] into the second equation:
[tex]\[
2(5 - 2y) + y = 16
\][/tex]
Step 3: Simplify the substituted equation:
[tex]\[
10 - 4y + y = 16 \implies 10 - 3y = 16
\][/tex]
Step 4: Solve for [tex]\( y \)[/tex]:
[tex]\[
-3y = 16 - 10 \implies -3y = 6 \implies y = -2
\][/tex]
Step 5: Substitute [tex]\( y = -2 \)[/tex] back into the expression for [tex]\( x \)[/tex]:
[tex]\[
x = 5 - 2(-2) = 5 + 4 = 9
\][/tex]
Step 6: Compute [tex]\( x + y \)[/tex]:
[tex]\[
x + y = 9 + (-2) = 7
\][/tex]
Thus, [tex]\( x + y \)[/tex] equals 7, so the correct answer is:
[tex]\[
\boxed{\mathbf{7}}
\][/tex]
