Elizabeth, Heather, and Erin share the same birthday. They’ve decided to join together to have one big birthday party. The event location says it will cost $70 for the rental of the space and $15 per guest. Each girl invites x number of guests, including themselves in their guest count.

The girls have each written a different correct expression to represent the total cost of the party. EXPLAIN how each of the 3 expressions below represents the total cost.
(a) Elizabeth’s expression: 15x + 15x + 15x + 70

(b) Heather’s expression: 15(x + x + x) + 70

(c) Erin’s expression: x(15 + 15 + 15) + 70

Elizabeth finds the cost of each girl's guests and adds the rental cost, for a result of the total cost.

Heather multiplies the cost per guest by the total number of guests, and adds the rental cost. This has the same result.

Erin multiplies the amount of guests each girl invites with the price per guest, taking the price three times since three girls invite guests. Then, she adds the rental cost.

Thus, all three equations are equal to the costs of the guests added to the rental cost, which is the full cost.