Problem 63 - College Algebra with Corequisite Support

Question

For the following exercises, use a system of linear equations with two variables and two equations to solve. The startup cost for a restaurant is $$\$ 120,000,$$ and each meal costs $$\$ 10$$ for the restaurant to make. If each meal is then sold for $$\$ 15,$$ after how many meals does the restaurant break even?

Step-by-Step Solution

Answer

The restaurant breaks even after selling 24,000 meals.

Step 1: Define the Variables

Let \( x \) represent the number of meals sold. Define the total cost \( C \) for producing \( x \) meals as \( C = 120000 + 10x \), where \( 120000 \) is the startup cost and \( 10x \) is the cost per meal. Define the total revenue \( R \) from selling \( x \) meals as \( R = 15x \).

Step 2: Set Up the System of Equations

To find the break-even point, set the total cost equal to the total revenue: \[ 120000 + 10x = 15x \] This equation represents the condition for breaking even, where costs equal revenue.

Step 3: Solve for \( x \)

Rearrange the equation \( 120000 + 10x = 15x \) to isolate \( x \). Subtract \( 10x \) from both sides: \[ 120000 = 5x \] Then divide both sides by 5 to solve for \( x \): \[ x = \frac{120000}{5} = 24000 \]

Step 4: Interpret the Solution

The value \( x = 24000 \) means the restaurant needs to sell 24,000 meals to break even. At this point, the costs of production are fully covered by the revenue, and they neither make a profit nor a loss.

Key Concepts

Break-even Analysis

Break-even analysis is a financial calculation to determine when a business, product, or project will start to be profitable. This is crucial for businesses, like a restaurant, to know when their sales cover their costs. In our exercise, the break-even point refers to the number of meals the restaurant must sell to cover the startup and operational costs without making a loss or profit.

Performing break-even analysis involves understanding both fixed and variable costs. Fixed costs are those that do not change with the level of output, such as the initial startup costs of \(\\)120,000\(. Variable costs change with the level of production—in this case, \)\\(10\) per meal.

By selling each meal for \(\\)15$, we can calculate when total revenues equal total costs. This involves setting up a system of equations to find the point where producing and selling meals will finally make the venture self-sustaining. Therefore, understanding break-even analysis helps businesses determine the minimum output needed to avoid losses.

Cost-Revenue Analysis

Cost-revenue analysis in business helps evaluate the financial performance over time by comparing costs and revenues. In the context of the restaurant, the exercise illustrates how one determines the relationship between money spent on producing meals and money earned from selling them.

To perform cost-revenue analysis, you start by defining:

• **Total Cost (C):** The sum of all costs to produce \(x\) meals. Here, it is expressed as \( C = 120,000 + 10x \), including the fixed startup cost and the variable cost per meal.
• **Total Revenue (R):** The income from selling \(x\) meals. In this case, it's \( R = 15x \), calculated by the price per meal times the number sold.

Understanding cost-revenue analysis enables businesses to decide pricing strategies, forecast profits, or make adjustments to cost structures. It's critical for ensuring that the pricing covers not just the current costs but also any future expenses and desired profit margins.

Linear Equations with Two Variables

Linear equations with two variables form the foundation for many business calculations, including break-even and cost-revenue analysis. These equations are in the form \( ax + by = c \), where \(a\), \(b\), and \(c\) are constants. In our restaurant problem, we have to solve such equations to find out how many units (meals) are needed to break even.

The equation set by our exercise \( 120000 + 10x = 15x \) is an example. Each side of the equation represents a different aspect of the problem; one side shows the cost equation, and the other shows the revenue equation. Solving this for \(x\) tells you the specific point (in terms of meal count) at which the cost equals the revenue.

By manipulating these equations, one sees clearly how changing prices or costs affect the break-even point. This not only helps in education but also prepares for real-world business challenges by enhancing analytical skills.