Evaluating Expressions: A Step-by-Step Guide

by Andrew McMorgan 45 views

Hey Plastik Magazine readers! Let's dive into a cool math problem that's all about checking if two expressions are the same. We'll be using a value for 'x' to see if everything lines up. It's like a math detective game, and we're the investigators! Our guide for today, Giovanni, is trying to figure out whether two expressions are equivalent.

The Expressions and the Question

So, here's the deal: we've got two expressions. The first one is $-4x - 8$ and the second one is $-2(x + 1) - 2(x + 3)$. Giovanni wants to know if these two are actually the same, or if they just look similar. To do this, he's going to replace 'x' with the number 3 in both expressions. What we're trying to figure out is: what do these expressions equal when x equals 3? This kind of problem is fundamental to algebra, and understanding it helps us with all sorts of other math stuff later on. It's like learning the ABCs before you read a novel – you gotta know the basics!

Let's break down the expressions and understand what the question is asking. We're essentially substituting a numerical value (in this case, 3) for the variable 'x' in each expression. After we do that, we'll simplify each expression using the rules of arithmetic (like following the order of operations: parentheses, exponents, multiplication and division, and addition and subtraction). The end result will be two numbers. Our goal is to find those two numbers and check which of the answer choices corresponds to those calculated values. Are you ready to start this math quest? Let's go!

Understanding the Goal

Our primary goal is to determine the numerical values of the two given algebraic expressions by substituting the value of 'x' with 3. This process involves a combination of substitution, and the application of arithmetic operations. The expressions are: $-4x - 8$ and $-2(x + 1) - 2(x + 3)$. Essentially, the aim of this question is to assess whether these two expressions are equal when x = 3. We will find this out by computing each expression separately and comparing the results with the options provided. Understanding the concept of algebraic equivalence is crucial here. Equivalent expressions yield the same results for all values of the variable.

Solving the First Expression: $-4x - 8$

Alright, let's start with the first expression: $-4x - 8$. This one's pretty straightforward. We're going to replace the 'x' with the number 3. So, the expression becomes: $-4 * 3 - 8$. Now, we just need to do the math. First, we multiply -4 by 3, which gives us -12. Then, we subtract 8 from -12. So, -12 - 8 equals -20. Got it? That means when x = 3, the first expression equals -20. Not too shabby, right? This step illustrates a fundamental concept in algebra: the process of substitution. Substitution involves replacing a variable with a specific value; in this case, we replaced 'x' with the number 3. This seemingly simple action transforms an algebraic expression into a numerical value that is, -20. Remember the order of operations? In this expression, multiplication comes before subtraction. So we first multiply -4 and 3 to get -12. Then we subtract 8 from -12, resulting in -20.

Step-by-step calculation

  1. Substitution: Replace 'x' with 3.

    βˆ’4(3)βˆ’8-4(3) - 8

  2. Multiplication: Multiply -4 by 3.

    βˆ’12βˆ’8-12 - 8

  3. Subtraction: Subtract 8 from -12.

    βˆ’20-20

Solving the Second Expression: $-2(x + 1) - 2(x + 3)$

Now, let's move on to the second expression: $-2(x + 1) - 2(x + 3)$. This one has a few more steps, but don't worry, we can totally handle it! Again, we're going to replace 'x' with 3. This gives us: $-2(3 + 1) - 2(3 + 3)$. Now, according to the order of operations, we need to deal with what's inside the parentheses first. So, 3 + 1 equals 4, and 3 + 3 equals 6. That changes the expression to: $-2(4) - 2(6)$. Next, we multiply: -2 times 4 equals -8, and -2 times 6 equals -12. So, the expression becomes: $-8 - 12$. Finally, we subtract: -8 - 12 equals -20. So, when x = 3, this expression also equals -20! You see, the steps might seem complicated, but breaking them down, one by one, makes the entire process way more manageable. The key concept at play here is the distributive property. It makes the simplification of the second expression easier. The distributive property allows us to multiply a term outside the parentheses with each term inside the parentheses. In each case, we first substitute x = 3 and then apply the distributive property.

Step-by-step calculation

  1. Substitution: Replace 'x' with 3.

    βˆ’2(3+1)βˆ’2(3+3)-2(3 + 1) - 2(3 + 3)

  2. Simplify inside parentheses:

    βˆ’2(4)βˆ’2(6)-2(4) - 2(6)

  3. Multiplication:

    βˆ’8βˆ’12-8 - 12

  4. Subtraction:

    βˆ’20-20

Comparing the Results

We did it, guys! We figured out the value of each expression when x = 3. The first expression, $-4x - 8$, equals -20. The second expression, $-2(x + 1) - 2(x + 3)$, also equals -20. Now, let's look at the answer choices. We're looking for an option that has -20 and -20 as the results. By comparing our findings with the options, we can confidently pick the correct answer. The important takeaway is that both expressions yield the same result, thus, they are equivalent for the given value of x. When comparing both results, we found the final answer. Now, let's select the correct option. The correct answer choice is the one that lists -20 for both expressions. The correct answer will indicate that both expressions are equal when x = 3. Understanding this is critical for algebraic equivalence.

The Final Answer

After evaluating both expressions, the correct option should be the one that gives -20 and -20. Among the choices, the correct answer is the option that matches our calculated values for both expressions when x equals 3.

The Correct Answer

Based on our calculations: when x = 3, both expressions equal -20. Therefore, the correct answer is the one that shows -20 for both expressions. Looking back at our options, we can now confidently select the correct one. Therefore, the correct answer is:

  • A. -20 and -20

Conclusion

Awesome work, everyone! We've successfully evaluated two algebraic expressions by substituting a value for a variable and simplifying them. We found that both expressions are equivalent when x = 3. This is an awesome example of how we use variables and numbers together in math. Keep practicing these skills, and you'll become a math whiz in no time. If you have any questions or want to try some more problems, let me know. Keep rocking, Plastik Magazine readers!