Verifying Algebraic Expressions: A Step-by-Step Guide

Hey guys! Ever wondered how to check if you've simplified an algebraic expression correctly? Today, we're diving into a super important skill in mathematics: verifying algebraic expressions. Specifically, we'll tackle a problem where Genevieve wants to verify if $-x$ is indeed the simplified form of the expression $\frac{1}{5}(5x-20) - \frac{1}{2}(4x-8)$. So, let's break it down and make sure we understand the process, alright?

Understanding the Problem

Before we jump into the solution, it's crucial to understand what we're trying to achieve. Verifying an algebraic expression means confirming that the simplified form is equivalent to the original expression. In Genevieve's case, she suspects that $\frac{1}{5}(5x-20) - \frac{1}{2}(4x-8)$ simplifies to $-x$. To verify this, we need a systematic approach. We need to ensure that every step we take is mathematically sound and leads us to the correct conclusion. The goal here is not just to get the answer but to understand why the answer is correct.

Think of it like this: you have a complex puzzle, and you think you've found the final piece. Verification is like making sure that piece actually fits and completes the puzzle perfectly. In algebra, this means ensuring that both sides of an equation are equal for all values of the variable (in this case, 'x'). Without this verification process, we might end up with incorrect simplifications, leading to mistakes in more complex problems down the road. That's why mastering this skill is so essential for anyone studying algebra and beyond!

Method 1: Simplify the Original Expression

One of the most straightforward ways to verify if $-x$ is the simplified expression is to actually simplify the original expression ourselves. This method involves carefully applying the order of operations and algebraic principles to transform the original expression into its simplest form. If, after simplification, we arrive at $-x$, then Genevieve's simplification is correct. If we end up with a different expression, we know there's been a mistake somewhere.

The process begins by distributing the constants outside the parentheses. This means multiplying $\frac{1}{5}$ by both terms inside the first set of parentheses $(5x - 20)$ and multiplying $\frac{1}{2}$ by both terms inside the second set of parentheses $(4x - 8)$. Remember to pay close attention to the signs! A negative sign in front of a fraction, like the $-\frac{1}{2}$ here, will affect the signs of the terms inside the parentheses.

After distributing, we combine like terms. Like terms are those that have the same variable raised to the same power (in this case, terms with 'x' and constant terms). We add or subtract the coefficients of these like terms to simplify the expression further. It’s essential to remember the rules of adding and subtracting integers and fractions during this step. A common mistake is to incorrectly combine terms that are not alike, so double-check that you’re only working with terms that have the same variable part.

Finally, we compare the simplified expression we obtained with $-x$. If they are the same, Genevieve’s verification is successful. If not, it indicates a need to re-evaluate the simplification steps to identify any errors. This method provides a clear, step-by-step way to confirm the correctness of the simplification, making it a valuable tool in algebra.

Step-by-Step Simplification

Let's walk through the simplification of the original expression, $\frac{1}{5}(5x-20) - \frac{1}{2}(4x-8)$, step by step, so we can see exactly how it works. This will not only help us verify Genevieve's result but also reinforce our understanding of algebraic simplification.

Step 1: Distribute the Constants

The first thing we need to do is distribute the fractions outside the parentheses. This means multiplying $\frac{1}{5}$ by each term inside the first set of parentheses and $-\frac{1}{2}$ by each term inside the second set. Let's break it down:

• $\frac{1}{5} \times 5x = x$
• $\frac{1}{5} \times -20 = -4$

So, $\frac{1}{5}(5x - 20)$ simplifies to $x - 4$.

Now, let's distribute the $-\frac{1}{2}$:

• $-\frac{1}{2} \times 4x = -2x$
• $-\frac{1}{2} \times -8 = 4$

So, $-\frac{1}{2}(4x - 8)$ simplifies to $-2x + 4$.

Step 2: Combine the Simplified Expressions

Now that we've distributed the constants, we have two simplified expressions: $x - 4$ and $-2x + 4$. We need to combine these:

$(x - 4) + (-2x + 4)$

Step 3: Combine Like Terms

Next, we combine like terms. Remember, like terms are terms that have the same variable raised to the same power. In this case, we have 'x' terms and constant terms.

Combine the 'x' terms: $x + (-2x) = -x$

Combine the constant terms: $-4 + 4 = 0$

Step 4: Final Simplified Expression

Putting it all together, our simplified expression is:

$-x + 0 = -x$

Conclusion of the Simplification

So, by simplifying the original expression step by step, we've arrived at $-x$. This means that Genevieve's claim that $\frac{1}{5}(5x-20) - \frac{1}{2}(4x-8)$ simplifies to $-x$ is indeed correct! Wasn't that cool how we could break down a complex expression into manageable steps?

Method 2: Substitute Values for x

Another fantastic way to verify if $-x$ is the simplified form of our expression is to substitute different values for 'x' into both the original expression and the simplified expression. If, for every value of 'x' we try, both expressions yield the same result, we can be highly confident that the simplification is correct.

The basic idea here is that if two expressions are equivalent, they should produce the same output for any given input. It's like having two different machines that are supposed to do the same job – if they're working correctly, they should give you the same product no matter what raw materials you feed them. In our case, the