Solve This! A Challenging Math Problem & Solution

Hey guys! Let's dive into a cool math problem today. We're going to break it down step-by-step so you can not only get the answer but also understand the why behind it. Math can seem intimidating, but I promise, with a little patience and the right approach, it can actually be pretty fun! We'll tackle a problem that might seem tricky at first glance, but I'll guide you through the solution process, making sure everything is crystal clear. So, grab your pencils, open your minds, and let's get started!

The Problem: A Real Brain Teaser

Okay, before we jump into the nitty-gritty, let’s lay out the problem we’re going to solve. Let's tackle a classic algebraic word problem. These types of problems are fantastic for honing your problem-solving skills because they force you to translate real-world scenarios into mathematical equations. Here it is:

The sum of three consecutive even integers is 78. Find the integers.

Now, I know what you might be thinking: “Ugh, word problems!” But trust me, we're going to break this down into manageable chunks. The key to solving these problems is to take your time, identify the unknowns, and translate the given information into mathematical expressions. Think of it as detective work – we have clues, and we need to piece them together to find the solution. We're not just looking for the answer here; we're building a foundation for tackling more complex problems in the future. This problem isn’t just about finding three numbers; it's about understanding how numbers relate to each other and how we can use algebraic principles to unravel those relationships. So, let's put on our thinking caps and dive deeper into the problem-solving process.

Understanding the Problem: Let's Break It Down

Before we even think about equations, let's really understand what the problem is asking. What are the key pieces of information we're given? We know a few important things:

• We're dealing with integers: Integers are whole numbers (no fractions or decimals). Examples: -3, -2, -1, 0, 1, 2, 3...
• They are consecutive: This means they follow each other in order (like 2, 3, 4 or 10, 11, 12).
• They are even: This means they are divisible by 2 (like 2, 4, 6 or 10, 12, 14).
• There are three of them: We need to find three even integers that fit our criteria.
• Their sum is 78: When we add these three integers together, we get 78.

This is a lot of information! But by breaking it down, we can see exactly what we need to find. We are essentially looking for a sequence of three numbers, and this sequence has very specific qualities that define it. The fact that these numbers are consecutive and even gives us a very important clue. It means there is a direct relationship between the numbers themselves. They are not random; they are part of a specific pattern. Recognizing this pattern is crucial because it allows us to express the unknowns in terms of each other, which will ultimately help us form our algebraic equation. We are essentially translating the words of the problem into a concrete mathematical framework.

Setting Up the Equation: Translating Words to Math

This is where the algebra comes in! We need to represent our unknowns (the three integers) with variables. Since they are consecutive even integers, we can represent them in relation to each other. Here's how:

• Let the first even integer be represented by x.
• Since the next consecutive even integer is 2 more than the previous one, the second integer is x + 2.
• Similarly, the third consecutive even integer is 2 more than the second, so it's x + 4.

Now, we know the sum of these three integers is 78. So, we can write the equation:

x + (x + 2) + (x + 4) = 78

This equation is the heart of the problem. It encapsulates all the information we have in a concise mathematical statement. Now that we've successfully translated the word problem into an equation, we've crossed a major hurdle. The equation is our roadmap; it provides a clear path toward the solution. Creating the equation correctly is the single most important step in solving this type of problem. A small error in this stage can lead to a completely wrong answer. So, take your time and double-check that your equation accurately reflects the relationships described in the problem.

Solving the Equation: Let's Get to Work!

Now comes the fun part – solving for x! Let's simplify the equation:

1. Combine like terms: 3x + 6 = 78
2. Subtract 6 from both sides: 3x = 72
3. Divide both sides by 3: x = 24

Great! We've found the value of x, which represents the first even integer. But we're not done yet! We need to find all three integers.

The process of solving the equation is where your algebraic skills really shine. Each step we take, from combining like terms to isolating the variable, is a fundamental algebraic operation. Understanding why we perform these operations is just as important as knowing how to perform them. We are essentially manipulating the equation to reveal the value of our unknown, x. The goal is to maintain the balance of the equation while simplifying it. Think of it as a mathematical dance – every move we make on one side, we must mirror on the other side to keep things in harmony. And remember, x is just the starting point. We've solved for one piece of the puzzle, but the final picture requires us to go back and use this value to find the other numbers in our sequence.

Finding the Integers: The Final Step

We know:

• The first integer is x = 24
• The second integer is x + 2 = 24 + 2 = 26
• The third integer is x + 4 = 24 + 4 = 28

So, the three consecutive even integers are 24, 26, and 28.

We've reached the final destination! We've successfully identified all three integers that satisfy the conditions of our problem. Each integer plays a critical role in fulfilling the initial requirements. However, the journey doesn't end here. A crucial step remains: verification. We must ensure that these integers not only fit the pattern of consecutive even numbers but also sum up to 78, as stated in the original problem. This act of checking our solution is a vital practice in mathematics and in any problem-solving endeavor. It reinforces our understanding and confirms the accuracy of our process.

Checking Our Work: Are We Sure?

Let's make sure our answer is correct! Add the integers together:

24 + 26 + 28 = 78

It works! Our solution is correct.

Always, always, always check your work! This is a crucial step in any math problem. It's easy to make a small mistake along the way, and checking your answer will catch those errors. In this case, we added our three integers together and confirmed that they do indeed sum up to 78, which is what the problem stated. This verification step gives us confidence that our solution is not only a set of three consecutive even numbers but also the correct set for this specific problem. It's like the final piece of the puzzle snapping into place, giving us a complete picture.

Conclusion: We Did It!

We successfully solved the problem! The three consecutive even integers that add up to 78 are 24, 26, and 28.

See? Word problems aren't so scary after all! By breaking them down step-by-step, understanding the information, setting up the equation, solving for the unknown, and checking our work, we can tackle any math challenge. The key is to approach each problem methodically and not to be afraid to ask for help or review the concepts if you get stuck. Math is a building process; each concept builds upon the previous one. So, by mastering the fundamentals, you'll be well-equipped to tackle more advanced topics. And remember, practice makes perfect! The more problems you solve, the more comfortable and confident you'll become. So, keep those pencils sharp, and let's keep exploring the wonderful world of mathematics together!

I hope this explanation helped you understand the process. Keep practicing, and you'll be a math whiz in no time!