Problem 53 - Algebra 1

Question

Use the table. It shows monthly expenses and income of a magazine for different numbers of subscribers. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \text { Number of subscribers } & 50 & 100 & 150 & 200 & 250 & 300 & 350 & 400 \\ \hline \text { Income } & \$ 75 & \$ 150 & \$ 225 & \$ 300 & \$ 375 & \$ 450 & \$ 525 & \$ 600 \\ \hline \text { Expenses } & \$ 150 & \$ 200 & \$ 250 & \$ 300 & \$ 350 & \$ 400 & \$ 450 & \$ 500 \\ \hline \end{array} $$ Look for patterns in the table. Write an equation that you can use to find how many subscribers the magazine needs for its income to equal its expenses.

Step-by-Step Solution

Answer

Solving the equation \(1.5*S = 100 + S/2\) yields the number of subscribers needed for the magazine's income to equal its expenses. The solution of this equation will be the answer.

Step 1: Understanding the Data

First, observe the values in the table. The income per subscriber seems to be increasing linearly by $1.50 for each added subscriber. Also notice that the expenses are rising in a linear fashion, increasing by $50 for every 50 additional subscribers.

Step 2: Setting Up the Equations

From the patterns observed in Step 1, the linear equations for income and expense can be written. For income: \(I = 1.5*S\), where \(S\) is the number of subscribers. For expenses: \(E = 100 + S/2\), where \(S\) is also the number of subscribers.

Step 3: Equating Income and Expense

The goal is to find the number of subscribers, \(S\), where the income is equal to the expense. So set the income equation equal to the expense equation and solve for \(S\), \(1.5*S = 100 + S/2\).

Key Concepts

Algebraic Patterns

Understanding algebraic patterns is akin to deciphering a secret code that governs how numbers interact within a system. When we look at the income and expenses in the given table, we notice that there's a consistent relationship between the number of subscribers a magazine has and the money it brings in or spends. This isn't random; it's an algebraic pattern. In this case, income increases by \(75 with every 50 additional subscribers, which means each subscriber contributes \)1.50 to the income. On the other side, expenses increase by $50 per every 50 subscribers. Recognizing these patterns allows us to model these relationships using linear equations, which predict income and expenses based on the number of subscribers.

Patterns in algebra are not just about spotting regularities. They are about translating behavior into equations that can be manipulated and used to make predictions. In the real world, this could help a business anticipate costs and revenue, proving crucial for its financial planning. Many students are intimidated by algebra because they see equations and variables, but it's really about understanding and applying these consistent patterns.

Solving Linear Systems

When we encounter two linear equations with a common variable, we have a 'system' of equations. Solving linear systems is about finding the values of these variables that make both equations true at the same time. In our exercise, we've identified two equations from the table: the income, represented by the equation I = 1.5*S, and the expenses, represented by E = 100 + S/2. To find the equilibrium point—the number of subscribers where income equals expenses—we set these equations equal to each other and solve for S.

Solving for the Number of Subscribers

By equating the income and expense equations:\[ 1.5*S = 100 + S/2 \]We clear the fractions and combine like terms to isolate S, which gives us the exact number of subscribers needed where the magazine's income will meet its expenses. This kind of analysis is not just a textbook exercise, but a practical tool businesses use to find their break-even point, set goals, and evaluate the impact of increasing or decreasing their customer base.

Income and Expense Analysis

Income and expense analysis is a critical aspect of both personal finance and business management. It is the process of examining how money is made and spent, to ensure stability or profitability. Our example provides a simplified version of this analysis by laying out the income and expenses associated with different numbers of subscribers. Through algebraic expressions, we crystalize the relationship between variables. If income doesn't match or exceed expenses, the venture isn't sustainable.

The importance of this analysis can't be overstated. For students grappling with the concepts, think of a magazine as a metaphor for your own finances or a future business. Your commitments (expenses) and your revenue (income) need careful balancing to ensure you don't operate at a loss. Whether you're managing a lemonade stand or a corporation, the principle remains the same—understanding and managing your financial inflows and outflows is paramount for success.