We are given that the total number of students in the auditorium is 50, where there are $2x$ boys and $y$ girls.

Step 1: Express the total number of students as an equation

$2x + y = 50$

After $(y-6)$ boys leave and $(2x-5)$ girls enter, the new number of boys and girls are:

• New number of boys = $2x - (y - 6) = 2x - y + 6$
• New number of girls = $y + (2x - 5) = y + 2x - 5$

Thus, the new total number of students: $(2x - y + 6) + (y + 2x - 5) = 4x + 1$

Step 2: Set up the probability equation

The probability of selecting a girl at random after the changes is given as:

$\frac{\text{New number of girls}}{\text{New total number of students}} = \frac{9}{13}$

Substituting values:

$\frac{y + 2x - 5}{4x + 1} = \frac{9}{13}$

Step 3: Solve for $x$ and $y$

Equation 1:

$2x + y = 50$

Equation 2 (Cross multiplying in the probability equation):

$13(y + 2x - 5) = 9(4x + 1)$

Expanding: $13y + 26x - 65 = 36x + 9$

Rearrange: $13y + 26x - 36x = 65 + 9$

$13y - 10x = 74$

Step 4: Solve the system of equations

1. $2x + y = 50$
2. $13y - 10x = 74$

From equation (1), express $y$ in terms of $x$:

$y = 50 - 2x$

Substituting in equation (2):

$13(50 - 2x) - 10x = 74$

$650 - 26x - 10x = 74$

$650 - 36x = 74$

$576 = 36x$

$x = 16$

Substituting $x = 16$ in $y = 50 - 2x$:

$y = 50 - 32 = 18$

Final Answer:

$\boxed{x = 16, y = 18}$

Would you like a detailed verification? Here are five related questions to explore:

1. How do you find the probability of selecting a student of a specific group?
2. What happens if a different number of students leave and enter?
3. How can probability equations be used in real-world applications?
4. What if the probability had been given as a different fraction?
5. How can we verify the correctness of the final solution?

Tip: Always double-check your algebraic manipulations to avoid calculation mistakes!