Let's solve the given problems step by step.

Part a)

We are tasked with finding the number of adults and children who attended the event. Here's how to approach it:

1. Define variables: Let:

   • $x$: number of adults
   • $y$: number of children

2. Formulate the equations:

   • The total number of seats is 200: $x + y = 200$
   • The total proceeds are $4200: $25x + 15y = 4200$

3. Solve the system of equations:

   From the first equation: $y = 200 - x$

   Substitute $y = 200 - x$ into the second equation: $25x + 15(200 - x) = 4200$

   Simplify: $25x + 3000 - 15x = 4200$ $10x = 1200$ $x = 120$

   Substitute $x = 120$ into $y = 200 - x$: $y = 200 - 120 = 80$

   Solution:

   • $x = 120$ adults
   • $y = 80$ children

Part b)

We are solving the system of linear equations: $3x - 2y = 10$ $5x + 6y = -58$

1. Use substitution or elimination. Let’s use elimination:

   • Multiply the first equation by 3 to align the $y$-coefficients: $9x - 6y = 30$
   • The second equation remains: $5x + 6y = -58$

2. Add the equations: $9x - 6y + 5x + 6y = 30 - 58$ $14x = -28$ $x = -2$

3. Substitute $x = -2$ into the first equation: $3(-2) - 2y = 10$ $-6 - 2y = 10$ $-2y = 16$ $y = -8$

   Solution:

   • $x = -2$
   • $y = -8$

Final Answers:

1. Part a: 120 adults and 80 children.
2. Part b: $x = -2$, $y = -8$.

Would you like more details or clarifications on these solutions? 😊

Here are 5 related questions for practice:

1. How would the solution change if the ticket price for children increased to $20?
2. What would happen if only 180 seats were filled instead of 200?
3. How do you solve linear systems using matrices or determinants?
4. Can you create your own word problem involving systems of equations?
5. What other methods could be used to solve Part b?

Tip: When solving systems of equations, double-check your substitution or elimination steps to avoid errors!