Math Problem: Auto Mechanic's Total Charges Equation

Hey guys, let's dive into a classic word problem that's super common in math classes, especially when you're just starting to get the hang of setting up equations. We're talking about a scenario involving an auto mechanic and their pricing structure. So, buckle up, because we're about to break down how to represent this situation mathematically. The core of this problem is understanding how different costs combine to form a total. In this case, we have a fixed diagnostic fee and a variable labor charge based on the time spent. This is a super common setup in real-world pricing, so mastering it will help you with all sorts of problems, not just this one. We need to figure out which equation correctly shows the relationship between the hours worked and the total cost. Let's get our hands dirty and figure out the right equation, shall we?

Understanding the Components of the Charge

Alright, let's break down the costs involved in this auto repair scenario. The auto mechanic has two main ways they charge you. First, there's a flat fee just to figure out what's wrong with your ride. This is the diagnostic charge, and it's set at $50. No matter how long it takes them to diagnose, or even if they diagnose it instantly, you're looking at that $50 charge just for the initial inspection. This is what we call a fixed cost because it doesn't change based on how much work is done afterwards. It's a one-time charge per visit for the diagnosis. Think of it as the entry fee to get your car looked at. Now, the second part of the charge is where the time factor comes in. The mechanic charges $25 for every hour they spend actually fixing the problem. This is the variable cost, and it directly depends on how long the repair takes. If they work for one hour, you pay $25. If they work for two hours, you pay $50. If they work for 'x' number of hours, you pay $25 multiplied by 'x'. So, we have a fixed amount ($50) and an amount that changes with time ($25 per hour). Our mission, should we choose to accept it, is to combine these two to create a single equation that tells us the total charges (let's call that 'y') based on the total number of hours worked (which we'll represent with 'x'). This is the essence of setting up linear equations from word problems, and it's a skill that's incredibly useful. We're essentially translating a real-world service into a mathematical model.

Building the Equation: From Words to Math

Now that we've dissected the costs, let's build the equation, guys! We're given two variables: x represents the total number of hours worked on the repair, and y represents the total charges. We need to find an equation that links y to x. Remember our breakdown? We have a fixed charge of $50 for the diagnosis. This $50 is applied regardless of how many hours are worked. So, it's a constant term that will be added to our total cost. Then, we have the hourly charge. The mechanic charges $25 per hour. If x is the number of hours, the total cost for the labor will be $25 multiplied by x, which we write as 25x. This part of the cost changes depending on the value of x. To get the total charges (y), we need to sum up the fixed diagnostic fee and the variable labor charges. So, we take the diagnostic fee ($50) and add the labor cost ($25x$). This gives us the equation: y = 50 + 25x. However, it's more conventional in mathematics, and often in how these problems are presented in multiple-choice options, to write the variable term first. So, we can rearrange this equation to y = 25x + 50. This equation precisely captures the situation: the total charge (y) is equal to the charge per hour (25) multiplied by the number of hours (x), plus the initial fixed diagnostic fee (50). This is a beautiful example of a linear equation in the form y = mx + b, where m is the slope (the rate of change, $25/hour) and b is the y-intercept (the starting value, $50). So, when you see these types of problems, always look for a fixed starting amount and a rate that changes with respect to a variable.

Evaluating the Options

We've done the heavy lifting and figured out what the equation should be: y = 25x + 50. Now, let's look at the options provided in the problem statement to see which one matches our derived equation. The options are:

A. y = 25x + 50 B. y = 50x + 25

Let's analyze each option based on our understanding of the problem.

Option A: y = 25x + 50

This equation suggests that the total charge (y) is calculated by taking the number of hours worked (x), multiplying it by $25 (the hourly rate), and then adding a fixed charge of $50 (the diagnostic fee). This perfectly aligns with our step-by-step derivation. The $25x$ part represents the cost of the labor, which varies with the hours worked, and the $+ 50$ represents the initial, constant diagnostic fee. This looks like our winner, folks!

Option B: y = 50x + 25

This equation implies that the total charge (y) is calculated by taking the number of hours worked (x), multiplying it by $50 (which would be the hourly rate), and then adding a fixed charge of $25 (which would be the diagnostic fee). This completely flips the numbers from our problem. In this scenario, the diagnostic fee would be $25, and the mechanic would charge $50 per hour. This is not what the problem describes. The problem clearly states a $50 diagnostic fee and a $25 hourly rate. Therefore, this option is incorrect.

So, after carefully analyzing the problem and the given options, we can confidently say that Option A is the correct equation that represents the auto mechanic's total charges. It's all about correctly identifying the fixed cost and the variable cost and assigning them to the right parts of the equation. Keep practicing these, and you'll be an equation-building pro in no time!