Let \( x \) represent the number of toppings added to the pizza.
The cost of a large cheese pizza at Peter's Pizzeria can be expressed as:
\[
C_P = 6.80 + 0.90x
\]
The cost of a large cheese pizza at Gavin's Pizzeria can be expressed as:
\[
C_G = 7.30 + 0.65x
\]
To find the number of toppings needed for the pizzas to cost the same, we set the two cost equations equal to each other:
\[
6.80 + 0.90x = 7.30 + 0.65x
\]
Now, we will isolate \( x \).
1. Subtract \( 0.65x \) from both sides:
\[
6.80 + 0.90x - 0.65x = 7.30
\]
\[
6.80 + 0.25x = 7.30
\]
2. Subtract \( 6.80 \) from both sides:
\[
0.25x = 7.30 - 6.80
\]
\[
0.25x = 0.50
\]
3. Divide both sides by \( 0.25 \):
\[
x = \frac{0.50}{0.25} = 2
\]
Thus, the number of toppings needed for the pizzas to cost the same is \( \boxed{2} \).