Calculating 37x + 37y - 37z When X + Y - Z = 5
Hey guys! Today, we're diving into a cool math problem that involves calculating the value of an expression given a specific condition. We're going to explore how to find the value of 37x + 37y - 37z when we know that x + y - z = 5. This type of problem is super common in algebra, and it's a great way to sharpen our math skills. Let's break it down step by step and make sure we understand every little detail. Trust me, it's not as complicated as it looks! We’ll use some basic algebraic principles to simplify the expression and arrive at the solution. So, grab your pencils and let's get started!
Understanding the Problem
So, what exactly are we trying to do here? We've got an expression, 37x + 37y - 37z, and we need to figure out its value. But we don't have specific values for x, y, and z. Instead, we have a relationship between them: x + y - z = 5. The key here is to use this relationship to simplify our expression and find a numerical value. Think of it like a puzzle where we have to rearrange the pieces to fit the solution. We need to connect the given equation with the expression we want to evaluate. This involves a bit of algebraic manipulation, but don’t worry, we'll take it nice and slow. Understanding the problem is the first step, and once we've got that down, the rest is just applying the right tools. We’re basically trying to see how the given equation can help us simplify the expression so that we can find a straightforward answer. It’s like having a secret code, and we need to decipher it to unlock the solution. So let's dive deeper and see how we can make this connection!
Key Concepts to Remember
Before we jump into solving, let's quickly recap some key concepts that will help us along the way. First off, the distributive property is going to be our best friend here. It allows us to factor out common terms from an expression. Remember, the distributive property states that a(b + c) = ab + ac. We'll be using this in reverse to factor out a common factor from our expression. Next, substitution is another crucial technique. Since we know the value of x + y - z, we can substitute it into our simplified expression to find the final value. Think of substitution as replacing one thing with its equal. These two concepts are the bread and butter of solving algebraic problems like this one. They help us transform complicated expressions into simpler, more manageable forms. By mastering these techniques, you'll be able to tackle a wide range of math problems with confidence. So, keep these in mind as we move forward, and you'll see how they make the process much smoother. We're building our toolkit, and these are essential tools for any budding mathematician!
Step-by-Step Solution
Okay, let's get our hands dirty and solve this problem! Here’s how we’re going to tackle it step-by-step:
Step 1: Factoring Out the Common Term
Look closely at our expression: 37x + 37y - 37z. Notice anything similar in each term? That's right, 37 is a common factor! We can use the distributive property in reverse to factor out 37. This means we rewrite the expression as 37(x + y - z). Factoring out common terms is a super useful trick in algebra. It simplifies the expression and makes it easier to work with. By pulling out the 37, we’ve transformed our expression into a product of 37 and another expression that looks very familiar. This is a big step forward because it directly links our expression to the given condition. Think of it as finding the common ground between what we have and what we know. We’re setting the stage for the next step, where we’ll use the given information to find our final answer. So, factoring out the common term is not just a neat trick; it’s a strategic move that brings us closer to the solution.
Step 2: Substituting the Given Value
Now comes the fun part! We've factored our expression to 37(x + y - z), and we know that x + y - z = 5. What does this mean? We can substitute the value of x + y - z with 5 in our expression. So, 37(x + y - z) becomes 37 * 5. See how easy that was? Substitution is a powerful tool because it allows us to replace a complex part of an expression with a simpler value, making the calculation straightforward. In this case, we're replacing an algebraic expression with a number, which significantly simplifies our task. This step highlights the beauty of math – how connecting different pieces of information can lead to a simple solution. We're essentially plugging in the missing piece of the puzzle, and the picture is becoming much clearer. This substitution step is the bridge that connects our factored expression with our given equation, bringing us one step closer to the final answer. So, let's move on to the final calculation and wrap this up!
Step 3: Final Calculation
Alright, we're in the home stretch! We've simplified our expression to 37 * 5. Now, all that’s left is to multiply these two numbers together. If you do the math, 37 multiplied by 5 is 185. So, the value of 37x + 37y - 37z, given that x + y - z = 5, is 185. And there you have it! We've successfully solved the problem. This final calculation is the payoff for all our hard work. It’s the moment when everything comes together, and we see the numerical result of our algebraic manipulations. Think of it as the last piece of a jigsaw puzzle sliding perfectly into place. We started with a seemingly complex expression and, through careful factoring and substitution, arrived at a single, definitive answer. This final step underscores the importance of accuracy and attention to detail in math. A simple multiplication brings us to the conclusion, demonstrating the power of step-by-step problem-solving. So, let’s celebrate our victory and recap what we've learned!
Conclusion
Woohoo, we did it! We successfully calculated the value of 37x + 37y - 37z when x + y - z = 5. By factoring out the common term and substituting the given value, we found that the answer is 185. This problem might have seemed tricky at first, but by breaking it down into smaller, manageable steps, we made it super easy. Remember, guys, the key to solving these types of problems is to identify common factors, use substitution, and take it one step at a time. These strategies not only help in math but also in tackling any complex problem in life. Each step we took was a building block, and together they formed a solid solution. This experience highlights the importance of perseverance and methodical thinking. So, next time you encounter a similar problem, remember our step-by-step approach, and you'll be solving it like a pro in no time! Keep practicing, and you’ll become a math whiz in no time. Let’s keep the learning momentum going and explore more exciting math challenges!
Key Takeaways
Let's quickly recap the key takeaways from our math adventure today. First, we learned the power of factoring out common terms. This technique simplifies complex expressions and makes them easier to work with. Factoring is like tidying up a room – it organizes the elements and makes the overall picture clearer. Second, we mastered the art of substitution. By replacing a part of the expression with its known value, we transformed the problem into a simple calculation. Substitution is the bridge that connects different pieces of information, allowing us to see the solution more clearly. Finally, we emphasized the importance of step-by-step problem-solving. Breaking down a complex problem into smaller, manageable steps makes it less daunting and more approachable. This methodical approach is not just useful in math but also in various aspects of life. By keeping these key takeaways in mind, you’ll be well-equipped to tackle similar problems and boost your problem-solving skills. So, remember these tools, and you’ll be a math superstar in no time!