Solving A System Of Equations Company Commute Scenario
Hey guys! Let's dive into an interesting math problem that involves setting up and solving a system of equations. This kind of problem is super practical, especially in business and economics, where you often need to figure out relationships between different quantities. So, buckle up, and let's get started!
The Curious Case of Commuting Employees
Imagine this: the owner of a company noticed something quite intriguing. The number of employees who drive their own cars to work is six times the number who use public transportation. On top of that, the owner also knows that if 13 employees switched from driving their own cars to using public transportation, the number of employees using public transportation would then be double the number driving their own cars. Sounds like a puzzle, right? Our mission is to figure out how many employees are currently driving their own cars and how many are using public transportation.
This is a classic math problem that we can tackle using a system of equations. Trust me, it's not as scary as it sounds! We'll break it down step by step, making it super easy to follow. First, we need to define our variables. Let's use 'x' to represent the number of employees who drive their own cars and 'y' to represent the number of employees who use public transportation. Now, we can translate the information from the problem into mathematical equations. Remember, the key to solving word problems is to break them down into smaller, manageable parts.
Setting Up the Equations: The First Clue
The first piece of information we have is that the number of employees who drive their own cars (x) is six times the number who use public transportation (y). We can write this as a simple equation:
x = 6y
This equation tells us the initial relationship between the number of drivers and public transport users. It's like the first piece of our puzzle, giving us a direct connection between 'x' and 'y'. Now, let's think about the second part of the problem. What happens when some employees switch their mode of transportation?
The Second Clue: The Switcheroo
The problem states that if 13 employees switched from driving their own cars to using public transportation, the number of employees using public transportation would be double the number driving their own cars. This is a bit more complex, but we can still write an equation for it.
If 13 employees switch from driving to public transport, the number of drivers decreases by 13 (x - 13), and the number of public transport users increases by 13 (y + 13). After this switch, the number of public transport users is double the number of drivers. So, we can write the second equation as:
y + 13 = 2(x - 13)
This equation captures the new relationship between drivers and public transport users after the switch. Now we have two equations, which means we have a system of equations ready to be solved!
Solving the System: Time to Get Our Math On!
Now that we have our system of equations:
- x = 6y
- y + 13 = 2(x - 13)
We can use a method of our choice to solve it. There are a few ways to tackle this, such as substitution or elimination. For this problem, the substitution method seems like a great fit because the first equation already gives us 'x' in terms of 'y'. This makes it easy to plug that value into the second equation. Let's see how it works!
Substitution Method: Plugging in the Values
Since we know that x = 6y, we can substitute 6y for 'x' in the second equation. This means we replace 'x' in the second equation with '6y'. Here’s how it looks:
y + 13 = 2(6y - 13)
Now we have a single equation with just one variable, 'y'. This is fantastic because we can solve it directly. Let’s simplify and solve for 'y':
y + 13 = 12y - 26
Solving for 'y': Finding the Number of Public Transport Users
To solve for 'y', we need to get all the 'y' terms on one side of the equation and the constants on the other side. Let’s subtract 'y' from both sides and add 26 to both sides:
13 + 26 = 12y - y
This simplifies to:
39 = 11y
Now, we divide both sides by 11 to isolate 'y':
y = 39 / 11
Oops! We seem to have hit a snag. We got y = 39/11, which isn't a whole number. Since we're talking about the number of employees, we need a whole number solution. Let's backtrack and check our work to make sure we haven't made any mistakes. This is a crucial step in problem-solving – always double-check!
Double-Checking Our Work: Catching Potential Errors
Okay, let’s go back and review our equations and steps. Our equations were:
- x = 6y
- y + 13 = 2(x - 13)
We substituted x = 6y into the second equation:
y + 13 = 2(6y - 13)
Expanding this gives:
y + 13 = 12y - 26
Let's rearrange the terms again:
39 = 11y
Ah, I see a small mistake! When subtracting 'y' from 12y, we correctly got 11y, and when adding 26 to 13, we got 39. So far, so good. It seems like our math is correct up to this point. However, the division 39 / 11 still gives us a non-integer value, which is odd. This suggests we might have made an error in setting up our equations initially or in interpreting the problem statement. It's time to revisit the original problem and our initial assumptions.
Reinterpreting the Problem: A Fresh Look
Let's reread the problem statement carefully: "The number of employees who drive their own cars to work is six times the number who use public transportation. If 13 employees switched from driving their own cars to using public transportation, the number of employees using public transportation would then be double the number driving their own cars." Sometimes, the wording of the problem can be tricky, and it’s important to make sure we’ve captured all the information correctly.
Our equations:
- x = 6y
- y + 13 = 2(x - 13)
Everything seems correct according to our initial interpretation. However, the non-integer result for 'y' is a red flag. It suggests there might be a slight misinterpretation or perhaps a typo in the problem statement itself (though we'll assume the problem is correctly stated for now). Let's think about the implications of the second equation more carefully.
Reevaluating the Second Equation: Ensuring Accuracy
The second equation, y + 13 = 2(x - 13), represents the situation after 13 employees switch from driving to public transportation. It states that the new number of public transport users (y + 13) is double the new number of drivers (x - 13). We need to ensure that this accurately reflects the scenario described in the problem.
Let's expand the second equation to better understand it:
y + 13 = 2x - 26
Now, let’s substitute x = 6y into this expanded equation:
y + 13 = 2(6y) - 26
y + 13 = 12y - 26
This is the same equation we had before, which led us to the non-integer solution. This persistent non-integer solution suggests that there may be no whole number solution that perfectly fits the conditions described in the problem. In real-world scenarios, this can happen due to various factors, such as approximations or slight inconsistencies in the data. However, since this is a mathematical problem, we expect a clean solution. This means we should consider whether we've missed a key aspect of the problem or made an assumption that isn't valid.
Trying a Different Approach: Elimination Method
Since the substitution method isn't giving us a clean answer, let’s try the elimination method. This can sometimes reveal inconsistencies or provide a different perspective on the problem. Our equations are:
- x = 6y
- y + 13 = 2(x - 13)
First, let’s rewrite the second equation in a more standard form:
y + 13 = 2x - 26
Rearranging gives:
2x - y = 39
Now our system of equations looks like this:
- x - 6y = 0 (rewriting the first equation)
- 2x - y = 39
To use the elimination method, we need to make the coefficients of either 'x' or 'y' the same (but with opposite signs) in both equations. Let’s eliminate 'x'. Multiply the first equation by -2:
-2(x - 6y) = -2(0)
This gives us:
-2x + 12y = 0
Now we have:
- -2x + 12y = 0
- 2x - y = 39
Add the two equations together:
(-2x + 12y) + (2x - y) = 0 + 39
This simplifies to:
11y = 39
Again, we get y = 39/11, which is not a whole number. This consistent result, regardless of the method we use, strongly suggests that there is no integer solution to this problem as it is stated.
Conclusion: Unraveling the Mystery
We've taken a deep dive into this problem, setting up a system of equations and trying both the substitution and elimination methods. Despite our best efforts, we keep arriving at a non-integer solution for the number of employees using public transportation. This outcome is quite telling. It implies that the conditions described in the problem might be inconsistent, or there might be no whole number solution that satisfies both conditions simultaneously. In real-world scenarios, this could mean that there’s a slight error in the data or that the situation is more complex than initially described.
So, what have we learned? First, setting up and solving systems of equations is a powerful tool for tackling problems in various fields. Second, it’s crucial to double-check our work and ensure we haven’t made any calculation errors. Third, and perhaps most importantly, sometimes problems don’t have neat, perfect solutions. This doesn't mean we've failed; it means we've learned something valuable about the problem itself and the real-world situations it represents. Keep practicing, guys, and you'll become math problem-solving pros in no time!
If you encounter a similar problem in the future, remember to revisit your assumptions, check your equations, and consider whether the problem as stated has a feasible solution. Math is not just about finding the right answer; it’s about the journey of problem-solving and the insights we gain along the way. Until next time, keep those equations balanced and your minds sharp!
