Problem 110 - A Graphical Approach to Precalculus with Limits

Question

Point 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. \(c=4 x+125\) \(R=9 x-200\)

Step-by-Step Solution

Answer

The break-even point is at 65 items, with costs and revenues both equal to $385.

Step 1: Set up the Equation

Identify the break-even condition, where cost equals revenue, i.e., set the equations for cost \(C\) and revenue \(R\) equal to each other. This gives us: \(4x + 125 = 9x - 200\).

Step 2: Solve for x

Rearrange the equation from Step 1 to collect all terms involving \(x\) on one side. Subtract \(4x\) from both sides to get: \(125 = 5x - 200\). Then add 200 to both sides to get: \(325 = 5x\). Finally, divide both sides by 5 to solve for \(x\): \(x = 65\).

Step 3: Substitute x back into C

Substitute \(x = 65\) back into the cost equation \(C = 4x + 125\) to find the cost at the break-even point. Thus, \(C = 4(65) + 125 = 260 + 125 = 385\).

Step 4: Substitute x back into R

Substitute \(x = 65\) back into the revenue equation \(R = 9x - 200\) to find the revenue at the break-even point. Thus, \(R = 9(65) - 200 = 585 - 200 = 385\).

Step 5: Verify the Break-even Point

Check that the calculated values for \(C\) and \(R\) are equal at \(x = 65\). Since both \(C\) and \(R\) equal 385, this confirms the solution is correct.

Key Concepts

Linear Equations

Linear equations are mathematical statements that describe a straight line when graphed on a coordinate plane. They are typically written in the form \(ax + b = 0\), where \(x\) is the variable, \(a\) is the coefficient of \(x\), and \(b\) is a constant. In the context of our problem, the cost and revenue functions are both expressed as linear equations:

• Cost: \(C = 4x + 125\)
• Revenue: \(R = 9x - 200\)

Linear equations like these are straightforward to work with because they represent simple relationships between variables. The variable \(x\) often represents a quantity such as the number of items produced or sold. Here, \(x\) is the number of items, and these equations determine cost and revenue associated with producing and selling \(x\) items.
These equations allow us to explore points of interest, such as the break-even point where cost equals revenue. Recognizing the characteristics of linear equations can help us visualize and solve problems much more efficiently.

Substitution Method

The substitution method is a technique used to solve systems of equations, where one equation is substituted into another. This method is very useful, especially when equations involve the same variables.
In our break-even problem, we're tasked with solving for \(x\) when \(C = R\), where \(C\) stands for cost and \(R\) for revenue. Here's how this method works:

• Set the two equations equal since you’re looking for the break-even point.
• We start with: \(4x + 125 = 9x - 200\).
• Rearrange terms to isolate \(x\). Move all terms containing \(x\) to one side of the equation.
• Solve for \(x\) by combining like terms and isolating \(x\). In this case, \(x = 65\).

Once you solve for \(x\), substitute it back into either the cost or revenue equation to find the corresponding values. This method simplifies the process and leads to a solution efficiently.

Precalculus

Precalculus serves as the foundation upon which calculus builds, focusing on concepts necessary to understand limits, derivatives, and integrals. While precalculus doesn’t delve into calculus topics themselves, it heavily involves algebra, geometry, and trigonometry, as well as an introduction to functions and systems of equations.
In this exercise, we're dealing with concepts found in precalculus, specifically linear equations and the substitution method, to analyze practical problems like the break-even point. The break-even point is key in business calculations and it’s where cost equals revenue. This is a critical application of algebra in a real-world scenario, bridging the gap between theoretical math and practical economics.
Mastering these concepts not only prepares you for more advanced studies in calculus but also enhances your problem-solving skills, allowing you to tackle complex financial problems with clarity and precision. By understanding precalculus thoroughly, you’ll find a smoother transition into the more abstract and rigorous world of calculus.