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When deciding between two or more options, it's essential to compare the associated costs effectively. Cost comparison involves analyzing different plans, prices, or strategies to determine the best choice based on financial factors. In the context of the given exercise, cost comparison is used to analyze two truck rental plans: Plan A and Plan B.

To compare them accurately, you need to understand the fixed and variable components of each plan.

• **Fixed Costs**: These are costs that remain constant regardless of usage, like the flat rate per week ( $75 for Plan A and $100 for Plan B).
• **Variable Costs**: These are costs that change based on usage, such as the cost per mile driven ($0.10 for Plan A and $0.05 for Plan B).

By identifying and comparing these elements, you can calculate the total cost for each plan over the same usage level, which allows for an informed decision about which plan is more cost-effective at different driving distances. The goal here is to find the breakeven point where both plans cost the same, which is determined by the number of miles driven.

Linear equations can simplify and solve real-world problems by modeling relationships in a straightforward mathematical form. In this exercise, linear equations represent the total cost of each rental plan based on miles driven.

• **Plan A Equation**: This is represented by the equation \(C_A = 75 + 0.10m\), reflecting a base cost plus a cost per mile.
• **Plan B Equation**: Similarly, for Plan B, it's \(C_B = 100 + 0.05m\), following the same pattern.

Using these equations, the process involves expressing costs as linear functions of the variable \(m\), which stands for miles driven. When you set these equations equal to each other, it indicates the point at which the costs of both plans are identical.

The beauty of linear equations is their simplicity and versatility. They help in visualizing how small changes in miles driven impact overall costs. By solving the equation \(75 + 0.10m = 100 + 0.05m\), you determine at what mileage both plans yield the same expenditure.

There are specific steps involved in methodically solving a problem, which makes the process easier and more systematic. Here, the focus is on solving for the breakeven mileage where two plans cost the same.

• **Understand the Problem**: Recognize what is being asked. Here, it is about finding equal costs between two plans.
• **Set Up Equations**: Translate the word problem into mathematical equations. For this problem, we use linear equations to represent each plan's cost.
• **Set Equations Equal**: Equate the two costs to find where they are the same.
• **Solve the Equation**: Simplify and solve for the unknown. For this scenario, solving \(0.10m - 0.05m = 25\) determines \(m\), the equal mileage.
• **Conclusion**: Interpret the results correctly. The solution means 500 miles is the point where plans are equally cost-effective.

Approaching problems with a structured method gives clarity and helps resolve even complex-looking problems. Solving for equal costs using these steps not only helps in this particular scenario but also builds skills applicable to many areas of math and daily life decision-making.