Part (a): Express the relationships as an algebraic system.

Let's define:

• Let $x$ represent the number of adult tickets sold.
• Let $y$ represent the number of senior tickets sold.

We know:

• The total number of people who attended is 331. So, we can write the equation for the total number of tickets sold: $x + y = 331$

• The total revenue from the tickets is $2517, and adult tickets are $9 while senior tickets are $7. Therefore, the equation for the total revenue is: $9x + 7y = 2517$

Thus, the system of equations is:

1. \quad & x + y = 331 \\ 2. \quad & 9x + 7y = 2517 \end{aligned}$$ --- ### Part (b): Solve the system using the substitution method. #### Step 1: Solve one equation for one variable. From the first equation $$x + y = 331$$, solve for $$x$$: $$x = 331 - y$$ #### Step 2: Substitute into the second equation. Now, substitute $$x = 331 - y$$ into the second equation $$9x + 7y = 2517$$: $$9(331 - y) + 7y = 2517$$ Simplify and solve for $$y$$: $$9(331) - 9y + 7y = 2517$$ $$2979 - 9y + 7y = 2517$$ $$2979 - 2y = 2517$$ Now, subtract 2979 from both sides: $$-2y = 2517 - 2979$$ $$-2y = -462$$ Now, divide both sides by -2: $$y = \frac{-462}{-2} = 231$$ #### Step 3: Substitute $$y = 231$$ back into the first equation. Now that we know $$y = 231$$, substitute it back into the equation $$x + y = 331$$: $$x + 231 = 331$$ Subtract 231 from both sides: $$x = 331 - 231 = 100$$ ### Final Answer: - The number of adult tickets sold is **100**. - The number of senior tickets sold is **231**. --- Let me know if you need further explanations or details! Here's some additional related questions: 1. How would you solve this system using the elimination method? 2. What would happen if the total number of tickets sold was different? 3. If the price of an adult ticket were reduced by $1, how would the system change? 4. How could you verify the solution through a graphical method? 5. What if the total receipts were increased to $3000, how would this affect the number of tickets sold? **Tip**: Always double-check the results by substituting the values of $$x$$ and $$y$$ back into both original equations to verify consistency!