Question

S. Parry PLANNING In Exercises 49 and 50 , you are planning a birthday party for your 8-year-old cousin. You can have the party at a pizza place for S8 per person plus \(\$ 30\) for favors and a small cake or at a taco place for \(\$ 12\) per person plus \(\$ 14\) for a large cake. How many children would you have to invite to the party for the cost to be the same for both places?

Short answer

In order for the cost to be the same at both the pizza place and the taco place, 4 children would need to be invited to the party.

Step-by-step solution

Formulate the Equations

Represent the cost of each location using an equation where 'x' represents the number of children. For the pizza place: \(C_{pizza} = 8x + 30\). For the taco place: \(C_{taco} = 12x + 14\).

Equate and Solve

Set the two equations equal to each other and solve for 'x'. By doing this, we are seeking the point where the total cost is the same. So, \(8x + 30 = 12x + 14\).

Simplify and Solve for 'x'

Rearrange terms to isolate 'x' in the equation. Subtract '8x' from both sides to get \(4x = 16\). Then, divide by '4' on both sides to solve for 'x', which yields \(x = 4\).