Problem 63 - College Algebra

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 Variables

Define two variables: Let \( x \) represent the number of meals prepared and sold, and \( C \) represent the cost incurred by the restaurant.

Step 2: Formulate the Profit Equation

Calculate the total revenue: Revenue from each meal is \( 15 \) dollars, so total revenue is \( 15x \). Calculate the total cost: The initial startup cost is \( \$ 120,000 \) plus \( 10 \) dollars per meal, leading to a total cost of \( 120,000 + 10x \). The restaurant breaks even when the total revenue equals the total cost; set the two equal: \[ 15x = 120,000 + 10x \].

Step 3: Solve the Equation

Rearrange the equation to solve for \( x \): Subtract \( 10x \) from both sides: \( 15x - 10x = 120,000 \), which simplifies to \( 5x = 120,000 \). Now divide both sides by 5: \[ x = \frac{120,000}{5} \].

Step 4: Calculate the Break-even Point

Perform the division to find \( x \): \( x = 24,000 \). This means the restaurant must sell 24,000 meals to break even.

Key Concepts

Linear Equations

A linear equation is a mathematical statement that represents a straight line when plotted on a graph. These equations are typically in the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants and \( x \) is the variable. Linear equations are fundamental in algebra because they are straightforward and help in solving real-world problems.
In the context of this exercise, we use linear equations to find out when the total revenue of the restaurant equals the total cost, which is known as the break-even point. By setting up the equation \( 15x = 120,000 + 10x \), we create a simple and direct way to find out how many meals need to be sold for the restaurant to cover all initial and ongoing costs.
Understanding linear equations is vital because they are used in various fields such as business, finance, and science for decision-making and predictions.

Break-even Point

The break-even point in business is the stage at which total revenues and total expenses are equal. It is an essential concept because it tells businesses how much they need to sell to cover all their costs.
In our exercise, the break-even point occurs when the revenue from selling meals \( 15x \) equals the sum of the startup cost \( 120,000 \) and the cost per meal \( 10x \).
By finding \( x \), which is 24,000 meals in this case, the restaurant knows exactly the number of meals needed to be sold in order to not operate at a loss.

• This concept helps in pricing strategies and financial forecasting.
• It allows businesses to assess their profitability and make informed decisions.

Breaking even is the first step towards achieving profitability and long-term success.

Problem Solving

Problem-solving involves breaking down a complex problem into manageable steps to find a solution. In this exercise, we used problem-solving techniques to determine how many meals a restaurant must sell to break even.
Here’s a brief overview of the steps used:

• Identify the problem and what you need to find, which is the break-even point.
• Formulate equations based on the relationships between different variables, like costs and revenues.
• Solve the equations step-by-step to find the solution.

Such a systematic approach is crucial in effectively tackling mathematical problems, ensuring logical and clear solutions. By practicing problem-solving, you enhance your analytical and critical thinking skills, which are valuable in everyday life and various professional fields.

Variables in Algebra

Variables are symbols used to represent unknown values in algebraic expressions and equations. They are fundamental in mathematics because they allow us to generalize and solve problems.
In our exercise, the variable \( x \) represents the number of meals sold. By defining variables, we can create equations that model the real-world situation, making it easier to analyze and solve.

• Variables allow flexibility in equations, letting you plug in different values to see how they affect the outcome.
• They help in forming relationships between different mathematical elements.

Understanding variables and how to manipulate them is pivotal for solving algebraic equations. It enables you to tackle a wide range of problems, from simple arithmetic to complex real-life scenarios.