To solve the given system of equations, we follow these steps:
[tex]\[
\begin{array}{l}
1. \, y = \frac{1}{3} x + 5 \\
2. \, y = 2x
\end{array}
\][/tex]
Step 1: Set the two expressions for [tex]\( y \)[/tex] equal to each other, since both equations describe [tex]\( y \)[/tex].
[tex]\[
\frac{1}{3} x + 5 = 2x
\][/tex]
Step 2: Solve the resulting equation for [tex]\( x \)[/tex].
First, eliminate the fraction by multiplying every term by 3:
[tex]\[
3 \left( \frac{1}{3} x \right) + 3 \cdot 5 = 3 \cdot 2x
\][/tex]
Simplifying:
[tex]\[
x + 15 = 6x
\][/tex]
Next, isolate [tex]\( x \)[/tex] by subtracting [tex]\( x \)[/tex] from both sides:
[tex]\[
15 = 6x - x
\][/tex]
[tex]\[
15 = 5x
\][/tex]
Finally, solve for [tex]\( x \)[/tex] by dividing both sides by 5:
[tex]\[
x = 3
\][/tex]
Step 3: Substitute [tex]\( x = 3 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. Using the second equation [tex]\( y = 2x \)[/tex]:
[tex]\[
y = 2 \cdot 3
\][/tex]
[tex]\[
y = 6
\][/tex]
Step 4: Find the value of [tex]\( x + y \)[/tex].
[tex]\[
x + y = 3 + 6 = 9
\][/tex]
Thus, the value of [tex]\( x + y \)[/tex] is:
[tex]\[
\boxed{9}
\][/tex]
