Algebraic Expression Errors: Find The Mistake & Correct Answer

Hey guys! Ever stared at a math problem and thought, "Wait, what just happened?" Well, you're in the right place! Today, we're diving into a classic algebraic expression evaluation mix-up that Gaira ran into. You know, those moments when you're sure you've got it, but the numbers just don't add up? We've all been there. So, let's break down Gaira's work step-by-step, figure out where things went sideways, and then nail the correct answer. Get ready to flex those math muscles because this is going to be fun!

The Original Problem: Evaluating $-3.8x + 6$ When $x = -7.5$

Alright, so Gaira was tasked with plugging in $x = -7.5$ into the algebraic expression $-3.8x + 6$. Sounds straightforward, right? You just substitute the value of $x$ and crunch the numbers. But as we'll see, even simple substitutions can sometimes lead to tricky results if you're not careful. The expression itself, $-3.8x + 6$, is a linear expression. This means it represents a straight line when graphed. The $-3.8$ is the coefficient of $x$, telling us how much the expression's value changes for every unit change in $x$. The $+6$ is the constant term, which is the value of the expression when $x$ is zero. When we're asked to evaluate it for a specific value of $x$, like $-7.5$, we're essentially finding the y-coordinate on that line at that particular x-coordinate. It’s like pinpointing a specific spot on a map. The challenge here often lies in dealing with negative numbers and decimals, which can sometimes throw people off. Gaira's attempt shows us exactly why paying attention to the details is super important in mathematics. This expression is fairly simple, but the presence of decimals and negative signs provides ample opportunity for errors, especially when multiplying or adding them. Let's see how Gaira tackled it.

Gaira's Work: A Closer Look

Here's what Gaira came up with:

-3.8(-7.5) + 6(-7.5)
28.5 + (-45)
(-16.5)

Now, let's dissect this. Gaira's first step involved substituting $x = -7.5$ into the expression. It looks like she correctly wrote down $-3.8(-7.5)$ for the first part. However, she also multiplied the $6$ by $-7.5$. This is where the first major red flag pops up, guys. The original expression is $-3.8x + 6$. When $x = -7.5$, we should substitute it only where $x$ appears. So, it should be $-3.8(-7.5) + 6$. There's no $x$ next to the $6$, so we shouldn't be multiplying $6$ by $-7.5$. This is a common mistake when people aren't super focused on the exact structure of the expression. They might see the $-7.5$ and just want to use it everywhere, or perhaps they misread the expression entirely. Let's assume Gaira was trying to apply the value of $x$ to the entire expression as if it were written as $(-3.8 + 6)x$, which is definitely not the case. Or maybe she was thinking of a different problem altogether. Regardless, the multiplication of $6 imes (-7.5)$ is an error based on the given expression. The second line, $28.5 + (-45)$, seems to stem from this initial misunderstanding. The calculation $-3.8 imes -7.5$ correctly yields $28.5$. Multiplying two negative numbers gives a positive result, and $3.8 imes 7.5$ does indeed equal $28.5$. So, that part is solid. The issue is the addition of $(-45)$. If Gaira had correctly multiplied $6 imes (-7.5)$, she would have gotten $-45$. So, it appears her error was twofold: first, she incorrectly included the $6$ in the multiplication with $x$, and second, she then proceeded with calculations based on that incorrect premise. The final result, $(-16.5)$, is the sum of $28.5$ and $-45$. And indeed, $28.5 - 45 = -16.5$. So, while the final arithmetic is correct based on her intermediate incorrect steps, the entire process is flawed because of the initial misinterpretation of the expression.

Identifying Gaira's Mistake

So, the key mistake Gaira made is incorrectly substituting the value of $x$. Specifically, she treated the expression as if it were $-3.8x + 6x$, or perhaps some other form where the $6$ was also multiplied by $x$. The original expression is $-3.8x + 6$. The variable $x$ is only present in the term $-3.8x$. Therefore, when we substitute $x = -7.5$, we should only replace the $x$ in that specific term. The constant term, $6$, remains unchanged. It's like having a recipe that calls for 2 cups of flour and 1 egg. If you decide to use a different type of flour (substitute), you only change the flour, not the egg! Gaira seems to have mistakenly multiplied the constant term $6$ by the value of $x$ as well. This is a crucial distinction in algebra. Terms are distinct parts of an expression separated by addition or subtraction. The term $-3.8x$ involves multiplication, while the term $6$ is just a constant. They are not treated the same way when substituting a variable's value. Think of it this way: if the expression was a sentence, and $x$ was a pronoun, you'd only replace the pronoun where it appears, not add it to other words in the sentence. So, Gaira's error wasn't in the arithmetic of multiplying negatives or adding positives; her error was in the setup of the calculation due to a misunderstanding of how algebraic expressions are structured and evaluated. It’s a common slip-up, especially when numbers get a bit messy with decimals and negatives.

The Correct Way to Evaluate the Expression

Now, let's get this right! We need to evaluate $-3.8x + 6$ when $x = -7.5$.

Step 1: Substitute the value of $x$.

Remember, we only replace $x$ where it appears. So, we substitute $-7.5$ for $x$ in the term $-3.8x$. The constant term $6$ stays as it is.

$$-3.8(-7.5) + 6$$

Step 2: Perform the multiplication.

We need to calculate $-3.8 imes -7.5$. Remember, a negative number multiplied by a negative number results in a positive number. Let's do the multiplication:

$3.8 imes 7.5$

We can break this down:

$3.8 imes 7 = 26.6$ $3.8 imes 0.5 = 1.9$

Adding these together: $26.6 + 1.9 = 28.5$.

So, $-3.8 imes -7.5 = 28.5$.

Our expression now looks like this:

$$28.5 + 6$$

Step 3: Perform the addition.

Finally, we add the results:

$$28.5 + 6 = 34.5$$

So, the correct answer when evaluating $-3.8x + 6$ for $x = -7.5$ is 34.5.

See the difference? By correctly substituting and following the order of operations, we arrive at a completely different and correct result. It’s a great reminder that precision in math is absolutely key, especially when you’re dealing with negative numbers and decimals. Keep practicing, guys, and you'll master these steps in no time!

Why Understanding Algebraic Structure Matters

It's really easy to get lost in the calculations when you're working with numbers, especially if they're decimals or negatives like in Gaira's problem. But what Gaira’s mistake highlights is the fundamental importance of understanding the structure of algebraic expressions. An expression like $-3.8x + 6$ is built from terms. In this case, we have two terms: $-3.8x$ and $+6$. The variable $x$ is only part of the first term. When we substitute a value for $x$, we are only changing the value of that specific variable within its term. The constant term, $6$, is independent of $x$. It’s a fixed value. This concept is crucial. If the expression were, for example, $(-3.8 + 6)x$, then Gaira’s substitution approach might have looked more logical (though still incorrect for the given expression). But as written, $-3.8x + 6$, the $6$ is separate. Think of it like ingredients in a recipe. If you're making cookies and the recipe calls for 2 cups of flour and 1 egg, and you decide to use whole wheat flour instead of all-purpose (a substitution for flour), you don't suddenly decide to substitute the egg with something else just because you changed the flour. You only change what the substitution applies to. In algebra, the substitution applies only to the variable itself. This careful attention to detail prevents errors. When you substitute $x = -7.5$ into $-3.8x$, you get $-3.8 imes (-7.5) = 28.5$. The $+6$ part is simply added on afterwards. So, $28.5 + 6 = 34.5$. This is the correct answer. Gaira’s mistake involved incorrectly applying the substitution to the constant term as well, as if the expression were $-3.8x + 6x$. This misunderstanding of terms and substitution is a common pitfall for students learning algebra. It reinforces the idea that math isn't just about crunching numbers; it's about understanding the rules and structures that govern those numbers. Paying close attention to parentheses, coefficients, and the separation of terms by addition and subtraction signs is essential for accurate mathematical reasoning. It’s the difference between getting the right answer and ending up with something completely off, like Gaira did. So, next time you’re evaluating an expression, pause for a second and check: where does this substitution actually belong in the expression? What are the distinct terms? This kind of thoughtful approach will save you a lot of headaches and ensure your mathematical journey is a successful one!

Conclusion: Learning from Mistakes

So, there you have it! Gaira's calculation, while demonstrating some correct arithmetic (like multiplying two negatives to get a positive), ultimately led to the wrong answer because of a fundamental misunderstanding of how to substitute a variable into an algebraic expression. The key takeaway is that you only substitute the value of the variable where the variable appears in the expression. The constant term, $6$, in $-3.8x + 6$, is unaffected by the value of $x$. Gaira incorrectly multiplied $6$ by $-7.5$, turning the expression into $-3.8x + 6x$, which is not what was asked. The correct evaluation involves calculating $-3.8(-7.5) + 6$, which correctly results in $28.5 + 6 = 34.5$. Mistakes are a natural part of learning, guys. The important thing is to understand why the mistake happened and how to correct it. By dissecting problems like this, we build a stronger foundation in mathematics. Keep practicing, stay curious, and don't be afraid to double-check your work. You’ve got this!