Simplifying Expressions A Step-by-Step Guide To $-5(3x - \frac{6}{7})$

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Hey there, math enthusiasts! Ever find yourself staring at an algebraic expression, feeling like it's a cryptic puzzle? Well, you're not alone! Let's break down one of those puzzles together. We're going to dive deep into simplifying expressions, focusing specifically on the expression $-5(3x - \frac{6}{7})$. Our mission? To figure out which of the given options is its true equivalent. So, buckle up, grab your thinking caps, and let's get started!

Delving into the Distributive Property

At the heart of simplifying expressions like this one lies a fundamental concept: the distributive property. Think of it as the golden rule of expression simplification. In simple terms, the distributive property tells us how to multiply a single term by a group of terms inside parentheses. It states that for any numbers a, b, and c:

a(b + c) = ab + ac

Now, let's apply this golden rule to our expression. We have $-5$ lurking outside the parentheses, and inside, we've got $(3x - \frac{6}{7})$. The distributive property says we need to multiply $-5$ by each term inside the parentheses. This means we'll be multiplying $-5$ by $3x$ and then $-5$ by $-\frac{6}{7}$.

Mastering the distributive property is like unlocking a superpower in algebra. It allows you to transform complex-looking expressions into simpler, more manageable forms. Without it, simplifying expressions would be like trying to solve a jigsaw puzzle with half the pieces missing. So, remember this rule, and you'll be well on your way to conquering algebraic challenges.

Let's walk through the process step by step. First, we multiply $-5$ by $3x$. This is straightforward: $-5 * 3x = -15x$. So far, so good! Now, let's tackle the second part: multiplying $-5$ by $-\frac{6}{7}$. Remember that multiplying two negative numbers results in a positive number. So, we have $-5 * -\frac{6}{7} = \frac{30}{7}$. Now we have all the pieces to simplify the expression. By applying the distributive property, we've broken down the original expression into manageable parts. This is a crucial step in solving any algebraic problem. Each part is simpler to handle, and together, they give us the solution we're looking for.

Applying the Distributive Property to Our Expression: A Step-by-Step Guide

Okay, let's roll up our sleeves and apply the distributive property to our specific problem: $-5(3x - \frac{6}{7})$. We'll break it down into bite-sized pieces so you can see exactly how it works.

Step 1: Multiply -5 by 3x

This is the first part of our distributive journey. We're taking the term outside the parentheses, $-5$, and multiplying it by the first term inside, which is $3x$. It's like saying, "Hey, $-5$, go visit $3x$!" When we do the math, we get:

$-5 * 3x = -15x$

So, the first part of our simplified expression is $-15x$. We're off to a good start!

Step 2: Multiply -5 by -6/7

Now, let's move on to the second part. We're still using that $-5$ from outside the parentheses, but this time, it's going to visit $-\frac{6}{7}$. Remember that multiplying two negative numbers gives us a positive result. So, this part is going to end up being positive. Let's do the math:

$-5 * -\frac{6}{7} = \frac{30}{7}$

So, the second part of our simplified expression is $+\frac{30}{7}$.

Step 3: Combine the Results

We've done the hard work! Now, we just need to put the pieces together. We found that $-5 * 3x = -15x$ and $-5 * -\frac{6}{7} = \frac{30}{7}$. So, we combine these two results:

$-15x + \frac{30}{7}$

And there you have it! By carefully applying the distributive property, we've transformed the original expression into its simplified form.

Breaking down the process into steps like this can make even the trickiest algebraic problems feel manageable. Each step is a small, achievable goal, and when you put them together, you've conquered the whole problem.

Identifying the Equivalent Expression: The Final Showdown

Alright, we've done the heavy lifting! We've simplified the expression $-5(3x - \frac{6}{7})$ using the distributive property, and we arrived at $-15x + \frac{30}{7}$. Now comes the fun part: comparing our simplified expression to the given options to find the match.

Let's remind ourselves of the options we were given:

• $-15x - \frac{30}{7}$
• $-15x - \frac{6}{7}$
• $-15x + \frac{6}{7}$
• $-15x + \frac{30}{7}$

Take a close look at each option. Which one looks exactly like the expression we simplified? It's like a mathematical version of "spot the difference!"

If you've been following along, the answer should jump out at you. The equivalent expression is:

$-15x + \frac{30}{7}$

High five! We did it! By carefully applying the distributive property and simplifying the expression, we were able to correctly identify the equivalent expression from the list. This is the kind of problem-solving prowess that makes math fun and rewarding.

Common Pitfalls and How to Avoid Them

Now that we've successfully navigated this problem, let's talk about some common stumbling blocks that students often encounter when simplifying expressions. Being aware of these pitfalls can help you avoid making mistakes and boost your confidence.

Pitfall #1: Forgetting the Distributive Property

This is a big one! It's easy to get caught up in the details and forget to distribute the term outside the parentheses to every term inside. Remember, the distributive property is the key to unlocking these types of expressions.

How to Avoid It: Always double-check that you've multiplied the term outside the parentheses by each term inside. A quick mental checklist can save you from this error.

Pitfall #2: Sign Errors

Ah, the dreaded sign errors! These can creep in when you're dealing with negative numbers. Remember that multiplying two negatives results in a positive, and multiplying a negative and a positive results in a negative.

How to Avoid It: Pay extra attention to the signs! Write out each step clearly, and don't try to do too much in your head. It's better to be methodical and accurate than to rush and make a mistake.

Pitfall #3: Fraction Fumbles

Fractions can sometimes feel intimidating, but they don't have to be! Remember the basic rules of fraction multiplication: multiply the numerators (top numbers) and multiply the denominators (bottom numbers).

How to Avoid It: If you're not comfortable multiplying fractions in your head, write it out! There's no shame in taking your time and being careful. Also, remember that a whole number can be written as a fraction with a denominator of 1 (e.g., -5 = -5/1). This can make the multiplication process clearer.

Pitfall #4: Combining Unlike Terms

This is a classic mistake! You can only combine terms that have the same variable and exponent. For example, you can combine $-15x$ with $-2x$, but you can't combine $-15x$ with $\frac{30}{7}$ because one has a variable and the other doesn't.

How to Avoid It: Before you start combining terms, make sure they're like terms! Underline or circle like terms to help you visualize which ones can be combined.

By being aware of these common pitfalls and taking steps to avoid them, you'll be well on your way to mastering expression simplification. Remember, practice makes perfect! The more you work with these concepts, the more confident you'll become.

Why This Matters: The Real-World Relevance of Algebraic Expressions

Okay, we've conquered this expression simplification problem, but you might be wondering, "Why does this even matter? When am I ever going to use this in real life?" That's a fair question! Let's explore the real-world relevance of algebraic expressions.

Algebraic expressions aren't just abstract symbols and numbers floating in a mathematical void. They're actually powerful tools that can be used to model and solve problems in a wide variety of fields. Think of them as a secret language that can help you understand the world around you.

Business and Finance

Imagine you're running a small business. You need to calculate your profits, track your expenses, and forecast your future earnings. Algebraic expressions can be your best friend in these situations.

For example, let's say you sell handmade jewelry. You have a fixed cost for materials and a variable cost for labor. You can use an algebraic expression to represent your total cost, and then use that expression to calculate your profit based on the number of pieces you sell. This kind of analysis is crucial for making informed business decisions.

Science and Engineering

From physics to chemistry to engineering, algebraic expressions are used to describe and predict how things work. Scientists use equations to model everything from the motion of planets to the behavior of chemical reactions. Engineers use them to design bridges, buildings, and machines.

For instance, the famous equation E=mc² (energy equals mass times the speed of light squared) is an algebraic expression that describes the relationship between energy and mass. This equation is fundamental to our understanding of the universe.

Everyday Life

You might be surprised to learn that you use algebraic thinking in your everyday life, even if you don't realize it! When you're planning a budget, calculating a tip at a restaurant, or figuring out how long it will take to drive to your destination, you're using the same kinds of problem-solving skills that you use in algebra.

For example, if you're trying to figure out how much paint you need to cover a wall, you might use an algebraic expression to calculate the area of the wall and then divide that by the coverage of the paint. This is a practical application of algebraic thinking in a real-world scenario.

The ability to understand and manipulate algebraic expressions is a valuable skill that can open doors in many areas of life. It's not just about solving equations in a textbook; it's about developing a way of thinking that can help you solve problems in any situation. So, the next time you're simplifying an expression, remember that you're not just doing math; you're building a powerful tool for understanding the world.

Wrapping Up: You've Got This!

Wow, we've covered a lot of ground! We've dissected the expression $-5(3x - \frac{6}{7})$, mastered the distributive property, identified the equivalent expression, and even explored the real-world relevance of algebra. You should be feeling pretty awesome right now!

The key takeaway from this journey is that simplifying algebraic expressions doesn't have to be a daunting task. By breaking it down into manageable steps, understanding the underlying principles, and being aware of common pitfalls, you can conquer even the trickiest problems.

Remember the distributive property: it's your secret weapon for simplifying expressions with parentheses. Pay attention to signs, and don't be afraid to take your time and write out each step. Practice makes perfect, so keep working at it, and you'll see your skills improve.

And most importantly, remember that algebra is more than just symbols and numbers. It's a powerful tool for problem-solving and critical thinking. The skills you develop in algebra will serve you well in many areas of your life.

So, go forth and simplify! You've got this!