Let the number of toppings be represented by $x$.

Luigi’s Pizzeria:

The cost of a large pizza at Luigi’s is: $\text{Cost at Luigi's} = 6.50 + 0.50x$

Alex’s Pizza:

The cost of a large pizza at Alex’s is: $\text{Cost at Alex's} = 8.00 + 0.25x$

To find the number of toppings $x$ where the costs are the same, we set the two cost equations equal to each other: $6.50 + 0.50x = 8.00 + 0.25x$

Solving for $x$:

1. Subtract $0.25x$ from both sides: $6.50 + 0.25x = 8.00$
2. Subtract 6.50 from both sides: $0.25x = 1.50$
3. Divide both sides by 0.25: $x = \frac{1.50}{0.25} = 6$

Conclusion:

To make the total cost the same, you need to add 6 toppings.

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Would you like further details on how to solve similar types of problems, or do you have any questions?

Here are five related questions:

1. How would you set up an equation if the initial costs or the cost per topping were different for each pizzeria?
2. How can you graph the two cost functions and find the point of intersection?
3. What happens to the number of toppings required if the cost of a topping increases at one of the pizzerias?
4. How would the answer change if tax was included in the cost calculations?
5. Can you find the point where the costs of a medium pizza and large pizza at both pizzerias are the same?

Tip: Always write down the cost formulas before starting to solve, and make sure both sides of the equation are balanced when solving for unknowns.