Let's solve the system of equations:

### Equations:
1. \( x + y = 9 \)  \[ (1) \]
2. \( 10y + x = 3(10x + y) - 9 \)  \[ (2) \]

### Simplifying Equation (2):
Expand and simplify equation (2):

\[
10y + x = 30x + 3y - 9
\]

Subtract \( x \) and \( 3y \) from both sides:

\[
10y - 3y + x - x = 30x - x + 9 
\]

This simplifies to:

\[
7y = 29x - 9
\]

Now, solve for \( y \):

\[
y = \frac{29x - 9}{7} \] \[ (3) \]

### Substitute \( y \) from Equation (3) into Equation (1):

Substitute the expression for \( y \) from Equation (3) into Equation (1):

\[
x + \frac{29x - 9}{7} = 9
\]

Multiply the entire equation by 7 to eliminate the fraction:

\[
7x + (29x - 9) = 63
\]

Simplify:

\[
36x - 9 = 63
\]

Add 9 to both sides:

\[
36x = 72
\]

Divide by 36:

\[
x = 2
\]

### Substitute \( x = 2 \) into Equation (1):

Substitute \( x = 2 \) into Equation (1):

\[
2 + y = 9
\]

Solve for \( y \):

\[
y = 7
\]

### Final Answer:
The solution to the system of equations is:

\[
x = 2, \quad y = 7
\]

Would you like to explore any further details or have any questions? 

Here are 5 related questions you might find interesting:

1. What happens if you swap the coefficients of \(x\) and \(y\) in the first equation?
2. How would the solution change if you added a constant to both sides of the first equation?
3. What is the significance of the second equation being more complex than the first?
4. How would you solve the system if it were non-linear (e.g., \(x^2 + y^2 = 9\))?
5. What are the implications of different methods (e.g., substitution vs. elimination) for solving systems of equations?

**Tip:** When dealing with systems of equations, always consider simplifying complex expressions as early as possible to make the calculations easier.

Learn how to solve the system of equations x + y = 9 and 10y + x = 3(10x + y) - 9 with a detailed step-by-step solution. This problem involves algebraic manipulation and substitution methods for solving linear equations. Suitable for students in Grades 9-12.

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