Problem 96 - A Graphical Approach to College Algebra

Question

The break-even point for a company is the point where costs equal revenues. If both cost and revenue are expressed as linear equations, the break-even point is the solution of a linear system. In each exercise, \(C\) represents cost in dollars to produce x items, and R represents revenue in dollars from the sale of \(x\) items. Use the substitution method to find the break-even point in each case-that is, the point where \(C=R .\) Then find the value of \(C\) and \(R\) at that point. $$\begin{aligned}&C=4 x+125\\\&R=9 x-200\end{aligned}$$

Step-by-Step Solution

Answer

The break-even point is at 65 items, with cost and revenue both being $385.

Step 1: Understand the Problem

We need to find the break-even point where cost \(C\) equals revenue \(R\). The equations are given as \(C = 4x + 125\) and \(R = 9x - 200\). We will use the substitution method to find the value of \(x\) where \(C = R\).

Step 2: Set the Equations Equal

To find the break-even point, set the cost equation equal to the revenue equation: \[4x + 125 = 9x - 200\]

Step 3: Solve for x

Subtract \(4x\) from both sides of the equation to isolate the terms involving \(x\): \[125 = 5x - 200\]. Then, add 200 to both sides: \[325 = 5x\]. Finally, divide both sides by 5 to solve for \(x\): \[x = 65\].

Step 4: Verify the Break-even Point

Use the value of \(x = 65\) to find \(C\) and \(R\). Substitute \(x = 65\) into the cost equation: \[C = 4(65) + 125 = 260 + 125 = 385\]. Substitute \(x = 65\) into the revenue equation: \[R = 9(65) - 200 = 585 - 200 = 385\]. Since \(C = R = 385\), this confirms our solution.

Key Concepts

Linear Equations

Linear equations are a fundamental part of algebra and are used to express relationships between variables in a straight line when graphed on a coordinate plane. A linear equation can be written in various forms such as the standard form, slope-intercept form, and more. For our purpose today, we focus on equations that are expressed in slope-intercept form, where an equation looks like this:
\[ y = mx + b \]- Here, \( m \) represents the slope of the line.- The variable \( x \) is the independent variable.- \( b \) is the y-intercept, indicating where the line crosses the Y-axis.In the context of break-even analysis, both cost \( C \) and revenue \( R \) are represented as linear equations. For example, the cost equation is given by \( C = 4x + 125 \), indicating a slope of 4 and a y-intercept of 125. Similarly, the revenue equation \( R = 9x - 200 \) shows a slope of 9 and a y-intercept of -200. The break-even point is where these two lines intersect, meaning the equations are equal.

Substitution Method

The substitution method is a commonly used technique for solving systems of linear equations. It is particularly useful when you have one equation in terms of a single variable. The goal is to substitute the value or expression of one variable from one equation into the other equation, allowing us to work with just one variable. Let's see how it works:
1. Begin with one of the given equations. For example, suppose you have \( C = 4x + 125 \) and \( R = 9x - 200 \).
2. Since you need to find when \( C = R \), set the equations equal to each other: \[ 4x + 125 = 9x - 200 \]3. Solve for \( x \) by isolating it on one side. By subtracting \( 4x \) from both sides, you get:\[ 125 = 5x - 200 \]4. Add 200 to both sides to simplify further:\[ 325 = 5x \]5. Divide by 5 to find \( x \):\[ x = 65 \]This means 65 items need to be produced and sold to reach the break-even point, where costs equal revenues. The substitution method makes it simpler to manage equations and find solutions effectively.

Cost and Revenue Analysis

Cost and revenue analysis is crucial for businesses to determine their financial performance and to make informed decisions. Through this analysis, businesses can find the break-even point, which indicates when the costs incurred are exactly covered by the revenue generated. Understanding this concept helps businesses set realistic production and sales targets.
In our exercise, the cost is represented by the equation \( C = 4x + 125 \), which means each item costs \( 4 \) dollars to produce, plus a fixed cost of \( 125 \) dollars. This fixed cost could include expenses like rent, salaries, or equipment that don't change with the production level.
Likewise, the revenue equation \( R = 9x - 200 \) shows that each item brings a revenue of \( 9 \) dollars, with \( 200 \) dollars in some form of initial negative offset such as promotional expenses or discounts.
By finding the point where cost equals revenue, represented by the conditions \( C = R \) and solved using methods like substitution, a business can identify how many items must be sold to start making a profit. At the break-even point, both cost and revenue are \( 385 \) dollars, showing the critical balance point for profitability.