Problem 45 - Algebra 1

Question

A local computer center charges nonmembers \(\$ 5\) per session to use the media center. If you pay a membership fee of \(\$ 25,\) you pay only \(\$ 3\) per session. Write an equation that can help you decide whether to become a member. Then solve the equation and interpret the solution.

Step-by-Step Solution

Answer

If you plan on attending 13 or more sessions, it would be more cost-effective to become a member.

Step 1: Set Up the Equation

Let's denote the number of sessions that will be attended by 'x'. Then, we can write down two cost equations - one for non-member which will be \(5x\) (since each session costs \(\$5\)), and the other for member which would be \(25 + 3x\) (since membership costs \(\$25\) and each session costs \(\$3\)). The break-even point is the number of sessions 'x' at which the costs will be equal, that is, \(5x = 25 + 3x\).

Step 2: Simplify the Equation

In order to find this 'x', we need to simplify the equation. First, subtract \(3x\) from both sides: \(5x - 3x = 25 + 3x - 3x\). This gives: \(2x = 25\).

Step 3: Solve for 'x'

Next, to isolate 'x', divide both sides of the equation by '2'. This leads to \(x = 25 / 2 = 12.5\).

Step 4: Interpret the Solution

As 'x' is a number of sessions, and it can't be a fraction, we need to round the result to the nearest whole number. In reality, you can't attend half of a session, so you need to attend at least 13 sessions for the membership to be cost-effective.

Key Concepts

Algebraic Problem Solving

Understanding how to approach algebraic problems is fundamental in various fields, including business and economics. When faced with an algebraic task, such as deciding whether to become a member of a club based on costs, the strategy is straightforward. Start by defining the variables that will affect the outcome—in this case, the number of sessions ().

Next, establish equations that reflect the different scenarios. For non-membership, the equation is simply the number of sessions times the per-session cost. Membership, however, includes a fixed cost plus a reduced rate. Representing these different scenarios mathematically allows for clear comparison and leads to establishing the break-even point – the value of the variable where costs are equalized. Algebra here is the tool that eliminates guesswork and provides tangible results to inform decision-making.

Break-Even Point Analysis

Break-even point analysis is a crucial concept for both business operations and everyday decisions. It is the moment where costs and revenues are equal, meaning there's no net loss or gain. It's applied here to determine after how many sessions the initial extra cost of a membership at the media center is justified.

The equation for the break-even point was derived by setting the costs equal to each other. At this point, any additional sessions would result in increased savings for the member. This analytical tool supports rational economic decisions and can guide individuals or businesses in planning their investments, pricing their products, or in this example, choosing between membership and non-membership options.

Linear Equations

The equations we formed to solve the given problem are examples of linear equations, which are foundational elements of algebra. A linear equation is any equation that can be written in the form (Ax + B = C), where (A), (B), and (C) are constants and (x) is the variable. In these equations, the graph of the solutions is a straight line, hence the term 'linear'.

In our scenario, (5x) for non-members and (25 + 3x) for members are both linear equations. Solving for the break-even point involved basic algebraic manipulations—simplifying and solving for the variable. These steps are typical in working with linear equations and are powerful tools in various real-life applications, including this cost analysis scenario.