Simplifying The Algebraic Expression (-2x * 13 - 71) / -5x A Step-by-Step Guide

Hey guys! Ever stumbled upon a mathematical expression that looks like a jumbled mess of numbers and variables? Today, we're going to break down one such expression: (-2x * 13 - 71) / -5x. Don't worry, it's not as scary as it looks! We'll go through it step by step, making sure you understand each part and how it all comes together. So, grab your metaphorical calculators (or real ones, if you prefer!) and let's dive in!

Understanding the Order of Operations

Before we even think about the specific numbers and variables in our expression, it's crucial that we understand the order of operations. Think of it as the golden rule of math – if you don't follow it, you'll end up with the wrong answer. The order of operations is often remembered by the acronym PEMDAS, which stands for:

• Parentheses (and other grouping symbols)
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Keeping PEMDAS in mind will be our guiding light as we navigate through the expression. In our case, we'll first deal with the multiplication within the parentheses, then the subtraction, and finally the division. Ignoring this order would be like trying to build a house starting with the roof – it just won't work! Remember, PEMDAS is your friend! It ensures we tackle mathematical problems in the correct sequence, leading to accurate results. When you approach a complex equation, always take a moment to map out your strategy based on PEMDAS. This simple step can save you a lot of headaches and prevent common errors. So, let's keep this in the back of our minds as we start simplifying the expression – it's the foundation of our entire process. Without a solid understanding of PEMDAS, even the simplest expressions can become confusing. Think of it as the grammar of mathematics; it provides the structure and rules for clear communication. So, let's make sure we're fluent in PEMDAS before moving on – it's the key to unlocking the mysteries of algebraic expressions. By consistently applying the order of operations, you'll develop a strong foundation for more advanced mathematical concepts and build confidence in your problem-solving abilities. Let’s move forward and unravel the layers of this expression, armed with the knowledge of PEMDAS as our trusted companion!

Breaking Down the Numerator: -2x * 13 - 71

The numerator is the top part of our fraction: -2x * 13 - 71. This is where we'll start our simplification journey. Following PEMDAS, we tackle the multiplication first. -2x * 13 is the same as -2 * 13 * x, which equals -26x. So, our numerator now looks like this: -26x - 71. This is as simplified as it can get for now, since -26x and -71 are not like terms (one has an 'x' and the other doesn't). We can't combine them. It's like trying to add apples and oranges – they're just different! Now, it's important to pause and appreciate what we've done. We've taken a chunk of the expression and simplified it using a fundamental mathematical principle. This step-by-step approach is the key to tackling complex problems. Think of it as solving a puzzle – you focus on fitting one piece at a time until the whole picture comes together. By breaking down the numerator, we've made the overall expression much less daunting. We've handled the multiplication, identified unlike terms, and set the stage for the next step. This methodical process not only helps us arrive at the correct answer but also builds our understanding of algebraic manipulation. So, let’s remember this strategy as we continue – break it down, simplify, and conquer! With each small victory, we're gaining momentum and confidence to tackle the rest of the expression. The numerator may seem simple now, but it’s a crucial piece of the puzzle. Now that we have -26x - 71, we are ready to move onto the next part of the equation and see how it all comes together. Keep in mind that each step we take is a building block, and with each simplified piece, we are getting closer to the final solution!

Understanding the Denominator: -5x

The denominator, the bottom part of our fraction, is -5x. In this case, it's already in its simplest form. There's nothing to multiply, divide, add, or subtract. It's just a coefficient (-5) multiplied by a variable (x). Sometimes, math gives you a break! It's important to recognize when a part of an expression is already simplified. This saves time and prevents unnecessary steps. In the grand scheme of things, identifying the simple parts allows us to focus our energy on the more complex sections. This is a valuable skill in problem-solving – knowing where to direct your efforts most effectively. In our expression, the denominator is a straightforward -5x, which means we can move on to the next phase with a clear understanding of what we're working with. It’s like having a solid foundation for a building; a simple, stable base allows us to construct more elaborate structures on top. So, let's appreciate the simplicity of -5x and use this momentum to tackle the next step. Remember, not every part of a mathematical problem needs extensive manipulation. Recognizing and accepting simplicity is just as important as knowing how to simplify complex terms. Let's carry this awareness forward as we continue our journey through the expression! With the numerator simplified and the denominator understood, we are now one step closer to unveiling the final form of the equation. Keep the momentum going, and let’s see what the next step holds!

Putting it Together: (-26x - 71) / -5x

Now we have the simplified numerator, -26x - 71, and the denominator, -5x. We can put them back together to form the fraction: (-26x - 71) / -5x. At this point, we could leave the expression as is, but we can also try to simplify it further. One way to do this is to split the fraction into two separate fractions: (-26x / -5x) + (-71 / -5x). This is a key technique in simplifying rational expressions. By breaking the fraction apart, we can often identify terms that can be simplified individually. In our first fraction, -26x / -5x, the x in the numerator and denominator cancels out, leaving us with -26 / -5. Since a negative divided by a negative is a positive, this simplifies to 26/5. The second fraction, -71 / -5x, also simplifies to a positive: 71 / 5x. So, our expression now looks like this: 26/5 + 71/5x. This is a more simplified form of the original expression. Remember, simplification is the name of the game! We're trying to make the expression as easy to understand and work with as possible. By splitting the fraction and canceling out common factors, we've taken a significant step towards this goal. This technique is a powerful tool in your mathematical arsenal, so make sure you understand it well. It’s like having a Swiss Army knife for math – it can come in handy in a variety of situations! So, let's take a moment to appreciate the progress we've made. We've gone from a seemingly complex expression to a much cleaner, more manageable form. And we did it by breaking it down, step by step, and applying some fundamental mathematical principles. This is the essence of problem-solving – taking something complicated and making it simple. Now, let's consider what we've achieved and think about any further simplifications that might be possible. Remember, the goal is to get the expression into its most elegant and understandable form. So, let's continue to explore the possibilities and see what other mathematical magic we can conjure up! With the fractions separated and simplified, we’re in a great position to put the final touches on this expression. Let’s keep pushing forward and see if we can uncover even more hidden simplicity!

The Final Simplified Expression: 26/5 + 71/5x

So, after all that simplifying, our expression (-2x * 13 - 71) / -5x becomes 26/5 + 71/5x. This is the simplified form of the expression. We've successfully navigated through the order of operations, combined like terms, and split fractions to arrive at this result. Awesome job, guys! Give yourselves a pat on the back! We've taken a complex-looking expression and transformed it into something much simpler and easier to understand. This is what math is all about – taking challenges head-on and finding elegant solutions. Remember, the key is to break things down into smaller, manageable steps and to apply the fundamental principles of mathematics. With practice and perseverance, you can tackle even the most daunting problems. This particular expression might seem like just one example, but the techniques we've used here can be applied to a wide range of algebraic problems. The ability to simplify expressions is a crucial skill in mathematics, and it will serve you well in future studies. So, let's take a moment to reflect on what we've learned. We've reinforced the importance of the order of operations, the power of combining like terms, and the art of splitting fractions. These are valuable tools in your mathematical toolkit, and you can use them to conquer any algebraic challenge that comes your way. As we conclude our journey through this expression, let's carry forward the confidence and skills we've gained. Math is not just about memorizing formulas and rules; it's about developing problem-solving abilities and a logical way of thinking. And with each expression we simplify, we're honing those skills and becoming more confident mathematicians. So, keep practicing, keep exploring, and keep simplifying! The world of mathematics is full of fascinating challenges, and with the right approach, you can overcome them all. Congratulations on mastering this expression, and let’s look forward to tackling even more exciting mathematical adventures in the future!

Real-World Applications

You might be thinking,