Simplifying Algebraic Expressions Finding Equivalent Forms

Hey guys! Let's dive into this math problem together and figure out which expression is equivalent to $(-4y - x) - (3y - 9x)$. This type of question often pops up in algebra, and it’s all about understanding how to combine like terms and handle those pesky negative signs. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into solving, let’s make sure we fully grasp what the question is asking. We have the expression $(-4y - x) - (3y - 9x)$, and our mission is to simplify it. This means we need to combine the terms that are similar—the y terms with the y terms, and the x terms with the x terms. The key here is to correctly distribute the negative sign in front of the second set of parentheses. This is where a lot of mistakes can happen, so we'll take it step by step to ensure we get it right.

Breaking Down the Expression

The expression we're working with is $(-4y - x) - (3y - 9x)$. Notice that we have two sets of parentheses and a subtraction sign between them. This subtraction sign is like a silent assassin – it changes the signs of everything inside the second set of parentheses. So, our first move is to distribute that negative sign. Think of it as multiplying each term inside the second parentheses by -1. This gives us:

$(-4y - x) - 1 * (3y - 9x)$

Now, let’s distribute the -1:

$(-4y - x) - 3y + 9x$

See how the sign of $3y$ changed from positive to negative, and the sign of $-9x$ changed from negative to positive? This is the most crucial step. If we mess this up, the whole solution goes off track. So, always double-check this part!

Combining Like Terms

Now that we've distributed the negative sign, we have a string of terms: $-4y - x - 3y + 9x$. The next step is to combine the like terms. Like terms are terms that have the same variable raised to the same power. In our case, the like terms are the y terms ($-4y$ and $-3y$) and the x terms ($-x$ and $+9x$).

Grouping Like Terms

It can be helpful to group the like terms together so we don't miss anything. Let's rewrite the expression, grouping the y terms and the x terms next to each other:

$-4y - 3y - x + 9x$

This makes it visually clearer which terms we need to combine. Now, let’s add the y terms together and the x terms together.

Adding the 'y' Terms

We have $-4y - 3y$. Think of this as owing 4 y’s and then owing another 3 y’s. In total, you owe 7 y’s. So, $-4y - 3y = -7y$.

Adding the 'x' Terms

Next, we have $-x + 9x$. This is like having 9 x’s and taking away 1 x. That leaves us with 8 x’s. So, $-x + 9x = 8x$.

The Simplified Expression

Now, let's put our simplified y and x terms together. We found that $-4y - 3y = -7y$ and $-x + 9x = 8x$. Combining these, we get:

$-7y + 8x$

So, the simplified expression is $-7y + 8x$. This is the final form, where we’ve combined all like terms and can't simplify any further.

Matching with the Options

Now that we have our simplified expression, $-7y + 8x$, we need to match it with the options given in the question:

A. $-7y - 8x$ B. $-7y + 8x$ C. $-y - 10x$ D. $y + 10x$

Looking at the options, we can see that option B, $-7y + 8x$, matches our simplified expression perfectly. So, the correct answer is B.

Common Mistakes to Avoid

When simplifying expressions like this, there are a few common pitfalls that students often stumble into. Being aware of these can help you avoid making the same mistakes.

Forgetting to Distribute the Negative Sign

As we mentioned earlier, the most common mistake is not correctly distributing the negative sign in front of the parentheses. Remember, that negative sign changes the sign of every term inside the parentheses. It’s like a mathematical ninja, silently flipping signs! Always double-check this step to ensure you've distributed correctly.

Combining Unlike Terms

Another frequent mistake is trying to combine terms that aren’t alike. You can only combine terms that have the same variable raised to the same power. For example, you can combine $-4y$ and $-3y$ because they both have y raised to the power of 1, but you can’t combine $-4y$ with $-x$ because they have different variables.

Arithmetic Errors

Simple arithmetic mistakes can also throw you off. Make sure you're adding and subtracting the coefficients correctly. It’s easy to make a small error, especially when dealing with negative numbers, so take your time and double-check your work.

Practice Makes Perfect

The best way to master simplifying expressions is to practice. The more you practice, the more comfortable you’ll become with the process, and the fewer mistakes you’ll make. Try working through similar problems, and don’t be afraid to make mistakes – that’s how we learn!

Example Practice Problems

Here are a couple of practice problems you can try:

1. Simplify: $(5a - 3b) - (2a + b)$
2. Simplify: $(-2x + 4y) - (-x - 2y)$

Work through these problems, paying close attention to distributing the negative sign and combining like terms. Check your answers to make sure you’re on the right track.

Conclusion

So, to wrap it up, the expression equivalent to $(-4y - x) - (3y - 9x)$ is $-7y + 8x$. We got there by distributing the negative sign, combining like terms, and carefully avoiding common mistakes. Remember, math is like building with Legos – each step builds on the previous one. Get the basics right, and you can build some pretty amazing things!

Keep practicing, stay curious, and you’ll become a math whiz in no time. You've got this, guys!