Problem 49 - Algebra 1

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?

Step-by-Step Solution

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 1: 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: \(C_{taco} = 12x + 14\).

Step 2: 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, \(8x + 30 = 12x + 14\).

Step 3: 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 \(x = 4\).

Key Concepts

Cost Comparison

When planning a party, comparing costs from different options can help in making an economically sound decision. In this case, we look at two different venues: a pizza place and a taco place. Each location has its own pricing strategy for hosting a birthday party.

• The pizza place charges $8 per child, plus a flat fee of $30 for favors and cake.
• Meanwhile, the taco place charges $12 per child, with a $14 fee for a larger cake.

The objective is to determine the number of children that need to be invited so the total cost at both venues is identical. By writing a mathematical equation for each location, you can directly compare costs and identify the number of children where both costs equalize. This is known as finding a breakeven point in cost comparison.

Equation Solving

Solving equations is a fundamental skill in algebra that allows you to find unknown values. In this scenario, we formulate equations to reflect the cost structures provided. For the pizza place, the equation is:\[ C_{pizza} = 8x + 30 \] And for the taco place:\[ C_{taco} = 12x + 14 \]By setting these two equations equal, you can solve for the number of children, \(x\), needed for the costs to be the same. The equation becomes:\[ 8x + 30 = 12x + 14 \]This step involves understanding that, when costs are equal, the expressions on either side of the equation are the same. Solving this equation will lead to discovering the specific value of \(x\).

Variable Isolation

Variable isolation is the process of rearranging an equation to solve for a specific variable. In our problem, we need to isolate \(x\) to find how many children need to attend for both venues to cost the same.Starting with the equation:\[ 8x + 30 = 12x + 14 \]Follow these steps:

• Subtract \(8x\) from both sides to simplify the equation: \(30 = 4x + 14\).
• Next, subtract \(14\) from both sides to further isolate the term with \(x\): \(16 = 4x\).
• Finally, divide both sides by \(4\) to completely isolate \(x\): \(x = 4\).

By isolating \(x\), you determine that inviting 4 children results in the same costs for both party venues. This is a crucial skill as it enables you to solve various real-world problems by finding unknown quantities.