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To solve the given system of equations, we'll find the values of \(x\) and \(y\) that satisfy both equations simultaneously. Here are the given equations:

1. \( y = \frac{1}{3}x + 5 \)
2. \( y = 2x \)

Since both expressions equal \(y\), we can set them equal to each other:

\[ \frac{1}{3}x + 5 = 2x \]

First, we'll eliminate the fraction by multiplying every term by 3:

\[ x + 15 = 6x \]

Next, we'll solve for \(x\) by isolating it on one side of the equation. Subtract \(x\) from both sides:

\[ 15 = 5x \]

Divide both sides by 5:

\[ x = 3 \]

Now that we have \(x\), we can substitute it back into either equation to find \(y\). Using the second equation:

\[ y = 2x \]
\[ y = 2(3) \]
\[ y = 6 \]

Therefore, the solution to the system of equations is \( (x, y) = (3, 6) \).

To find the value of \(y + x\):

\[ y + x = 6 + 3 = 9 \]

Thus, the value of \( y + x \) is \( \boxed{9} \).

Would you like any further details or have any questions?

Here are some more questions you might find helpful:
1. How do you solve systems of equations by substitution?
2. What are the methods to solve systems of linear equations?
3. Can you explain the elimination method for solving systems of equations?
4. How do you graphically solve a system of linear equations?
5. What are some common applications of systems of equations?

**Tip:** When solving systems of equations, always check your solutions by substituting them back into the original equations to ensure they satisfy both equations.

Learn how to solve a system of equations by finding values for x and y that satisfy both equations simultaneously. Detailed step-by-step solution included. Suitable for high school students preparing for exams.

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