Solve: Two Numbers, Difference Of 6, Sum Of 40

Introduction

Hey guys! Let's dive into a classic math problem that many students encounter: solving for two numbers when one exceeds the other by a certain amount, and their sum is known. This type of problem is a fantastic way to sharpen your algebraic skills and your ability to translate word problems into mathematical equations. In this article, we're going to break down a specific example step-by-step, ensuring you understand not just the how but also the why behind each step. We'll tackle the scenario where we need to find two numbers where one exceeds the other by 6, and their sum totals 40. So, grab your pencils and let’s get started!

At its core, this problem embodies the fundamental principles of algebra. We're presented with a verbal puzzle, and our mission is to convert it into a solvable mathematical form. The beauty of algebra lies in its ability to represent unknowns with variables, allowing us to manipulate these representations and ultimately uncover their values. Think of it like detective work – we're given clues, and we use logic and mathematical tools to solve the mystery. This specific problem is a common type that you'll see in algebra courses, so mastering it now will definitely pay off later. Plus, these kinds of problems aren't just confined to textbooks; they often pop up in real-world scenarios, like splitting costs or figuring out quantities. So, let’s unravel this numerical enigma and equip ourselves with valuable problem-solving skills!

Remember, the key to success with these problems is a blend of careful reading, clear thinking, and consistent practice. Don’t be afraid to take your time, re-read the problem, and break it down into smaller, more manageable chunks. As we move through the solution, we’ll emphasize the importance of each step and how it contributes to the overall answer. We'll explore how to define our variables, construct our equations, and systematically solve for the unknowns. By the end of this article, you'll not only be able to solve this particular problem but also approach similar problems with confidence and a clear strategy. So, let’s get ready to translate those words into numbers and unlock the secrets hidden within this mathematical puzzle!

Setting Up the Equations

Okay, the first crucial step in solving any word problem is to carefully read and understand what we're being asked to find. In our case, we're looking for two numbers. Let's call these numbers 'x' and 'y'. This is where the power of algebra comes in – we can use these variables to represent the unknowns we're trying to find. Now, let's translate the information given in the problem into mathematical equations. The problem tells us that one number exceeds the other by 6. This means that if we subtract the smaller number from the larger number, we should get 6. We can express this mathematically as: x = y + 6 or x - y = 6. It doesn't matter which variable you consider the larger one initially; the important thing is to be consistent throughout the problem.

The next piece of information is that the sum of the two numbers is 40. This is a pretty straightforward statement that we can directly translate into an equation: x + y = 40. So, now we have two equations: x = y + 6 and x + y = 40. This is a system of equations, and we have several methods at our disposal to solve it. One common method is substitution, where we solve one equation for one variable and then substitute that expression into the other equation. Another method is elimination, where we manipulate the equations so that when we add or subtract them, one of the variables cancels out. We'll use the substitution method in this case, as it feels quite natural given our first equation is already solved for x.

Think of setting up these equations as building the foundation for our solution. If our equations aren't accurate, the rest of our work will be for naught. It's like trying to build a house on shaky ground – it just won't stand! So, take your time, double-check your translations, and make sure your equations accurately reflect the relationships described in the problem. This meticulous approach will save you headaches down the road and set you up for success. Remember, practice makes perfect, so the more you work through these types of problems, the more comfortable you'll become with the process of translating words into the language of mathematics. Now that we have our foundation laid, let’s move on to the exciting part: solving for our unknowns!

Solving the System of Equations

Alright, now that we have our two equations – x = y + 6 and x + y = 40 – it's time to roll up our sleeves and solve for those variables. As mentioned earlier, we'll use the substitution method here. Since the first equation already has x isolated (x = y + 6), we can substitute this expression for x into the second equation. This gives us: (y + 6) + y = 40. See what we did there? We replaced the x in the second equation with the expression y + 6 from the first equation. This is a super powerful technique that allows us to reduce a two-variable problem into a single-variable problem, which is much easier to solve.

Now, let's simplify this new equation. We have (y + 6) + y = 40. Combining the 'y' terms, we get 2y + 6 = 40. Our goal now is to isolate 'y'. To do this, we first subtract 6 from both sides of the equation: 2y + 6 - 6 = 40 - 6, which simplifies to 2y = 34. Next, we divide both sides of the equation by 2: 2y / 2 = 34 / 2, which gives us y = 17. Awesome! We've found the value of one of our variables. But we're not done yet; we still need to find the value of x.

To find x, we can plug our value for y (y = 17) back into either of our original equations. It's generally easier to use the equation where x is already isolated, which is x = y + 6. So, substituting y = 17, we get x = 17 + 6, which simplifies to x = 23. And there you have it! We've solved for both x and y. We found that x = 23 and y = 17. Remember, the key to successfully solving systems of equations is to be systematic and careful with your algebraic manipulations. Double-check each step to avoid errors, and don’t hesitate to re-work the problem if you feel unsure. Now, let’s move on to the final and crucial step: verifying our solution.

Verifying the Solution

Okay, we've crunched the numbers and found our solutions: x = 23 and y = 17. But before we do a victory dance, it's super important to verify that our answers actually work. This is like the quality control step in any problem-solving process. Verifying our solution involves plugging our values for x and y back into the original equations to make sure they hold true. This helps us catch any potential errors we might have made along the way.

Let's start with the first equation, which stated that one number exceeds the other by 6. We can write this as x - y = 6. Plugging in our values, we get 23 - 17 = 6. Is this true? Yes, 6 = 6, so our solution works for the first equation. Now, let's check the second equation, which stated that the sum of the two numbers is 40. We can write this as x + y = 40. Plugging in our values, we get 23 + 17 = 40. Is this true? Yes, 40 = 40, so our solution works for the second equation as well.

Since our values for x and y satisfy both of the original conditions stated in the problem, we can confidently say that we've found the correct solution. High five! Guys, this step might seem like extra work, but it’s absolutely essential. Verifying our answers gives us peace of mind and helps us avoid silly mistakes that can cost us points on a test or in real-world applications. It also reinforces our understanding of the problem and the solution process. Think of it as the final seal of approval on our mathematical journey. Now that we've successfully navigated this problem from start to finish, let's take a moment to recap the key steps we followed.

Conclusion and Key Takeaways

Woohoo! We've successfully tackled this math problem, finding two numbers where one exceeds the other by 6 and their sum is 40. We found that the numbers are 23 and 17. But more importantly than just getting the right answer, we've walked through the process of problem-solving in a systematic way. Let's quickly recap the key steps we took, as these are applicable to many other math problems, and really, to problem-solving in general.

First, we carefully read and understood the problem. This involves identifying what we're being asked to find (in this case, two numbers) and noting the given relationships between them. This step is crucial because misinterpreting the problem can lead us down the wrong path from the get-go. Second, we translated the words into mathematical equations. This is where we use variables to represent the unknowns and express the relationships between them as equations. We came up with two equations: x = y + 6 and x + y = 40. Third, we solved the system of equations. We chose the substitution method, but there are other methods like elimination that could also be used. We substituted the expression for x from the first equation into the second equation, solved for y, and then plugged the value of y back into one of the equations to find x. Fourth, and this is a big one, we verified our solution. We plugged our values for x and y back into the original equations to make sure they held true. This step is our safety net, catching any errors we might have made along the way.

So, what are the key takeaways from this exercise? Well, first and foremost, practice is key! The more you work through these types of problems, the more comfortable and confident you'll become. Second, a systematic approach is essential. Break down the problem into manageable steps, and tackle each step one at a time. Third, don't underestimate the power of verification. Always, always, always check your work! Finally, remember that algebra is a powerful tool for solving real-world problems. The skills you're developing here aren't just for the classroom; they're applicable to many aspects of life. Keep practicing, keep asking questions, and keep challenging yourselves. You guys have got this!