Equivalent Expression Of 2(7x + 4) - 4(3 + 3x - 2)
Hey guys! Let's break down this math problem together. We're going to figure out which expression is the same as the one given: 2(7x + 4) - 4(3 + 3x - 2). It might look a bit intimidating at first, but don't worry, we'll take it step by step. Our main goal here is to simplify this expression by distributing the numbers outside the parentheses and then combining like terms. This is a fundamental concept in algebra, and mastering it will help you tackle more complex problems later on. So, grab your pencils, and let's dive in!
Understanding the Problem
Before we jump into solving, let's make sure we understand exactly what the question is asking. The prompt asks us to find the equivalent expression. What does that mean? An equivalent expression is simply another way of writing the same thing. It's like saying "hello" versus "hi"—they mean the same thing, but they look different. In this case, we need to simplify the given expression and see which of the answer choices matches our simplified version. This involves using the distributive property and combining like terms, both crucial skills in algebra.
Breaking Down the Expression
Okay, let's look closely at our expression: 2(7x + 4) - 4(3 + 3x - 2). We can see two main parts here, each involving parentheses. The first part is 2 multiplied by everything inside the first set of parentheses (7x + 4). The second part is -4 multiplied by everything inside the second set of parentheses (3 + 3x - 2). Notice that minus sign in front of the 4 – it's super important to include that negative when we distribute! This is a common area where students make mistakes, so keep an eye on those signs. Remember, accurate distribution is key to getting the correct answer. Let's move on to the next step and actually do the distribution.
Step-by-Step Solution
Now for the fun part – solving! We're going to tackle this problem step by step to make sure we don't miss anything. Remember, accuracy is more important than speed, especially in algebra. So, let's take our time and do it right.
1. Distribute the 2
First, we'll distribute the 2 in the first part of the expression: 2(7x + 4). This means we multiply 2 by both 7x and 4. So, 2 * 7x gives us 14x, and 2 * 4 gives us 8. Therefore, 2(7x + 4) becomes 14x + 8. Make sure you're comfortable with this process – it's the foundation for simplifying many algebraic expressions. Distribution is like sharing – the number outside the parentheses gets multiplied by each term inside. Got it? Great, let's move on to the next part!
2. Distribute the -4
Next up, we need to distribute the -4 in the second part of the expression: -4(3 + 3x - 2). This is where it's crucial to remember that negative sign! We're multiplying -4 by 3, 3x, and -2. So, -4 * 3 gives us -12, -4 * 3x gives us -12x, and -4 * -2 gives us +8. Notice how multiplying two negatives gives us a positive. This is a super important rule to remember in math. So, -4(3 + 3x - 2) becomes -12 - 12x + 8. We've now successfully distributed in both parts of the expression. Let's keep going!
3. Combine Like Terms
We're almost there! Now we need to combine like terms. This means we'll group together the terms with 'x' and the constant terms (the numbers without 'x'). Our expression now looks like this: 14x + 8 - 12 - 12x + 8. Let's rearrange the terms to group like terms together: 14x - 12x + 8 - 12 + 8. Now it's easier to see what we can combine. 14x minus 12x is 2x. 8 minus 12 is -4, and -4 plus 8 is 4. So, when we combine like terms, we get 2x + 4. See? It's not so scary when we break it down step by step. This is the simplified version of our original expression.
Identifying the Equivalent Expression
Alright, we've done the hard work of simplifying the expression. Now, we need to look at the answer choices and see which one matches our simplified expression, which is 2x + 4. This is the final step, and it's crucial to make sure we pick the correct answer.
Matching the Solution
Let's quickly recap the answer choices:
A. 4x - 4 B. 2x - 4 C. 2x + 4 D. 14x + 4
By comparing our simplified expression 2x + 4 with the answer choices, we can clearly see that option C, 2x + 4, is the correct answer. This is a perfect example of how breaking down a problem into smaller, manageable steps can lead us to the solution. We distributed, combined like terms, and now we've confidently identified the equivalent expression. High five!
Common Mistakes to Avoid
Before we wrap up, let's chat about some common mistakes people make when solving problems like this. Knowing these pitfalls can help you avoid them and boost your accuracy.
Sign Errors
One of the most frequent mistakes is messing up the signs, especially when distributing negative numbers. Remember, a negative times a negative is a positive, and a negative times a positive is a negative. Pay close attention when distributing negative numbers, like the -4 in our example. Double-checking your signs can save you from a lot of headaches.
Incorrect Distribution
Another common error is not distributing correctly. Make sure you multiply the number outside the parentheses by every term inside the parentheses. It's easy to forget one, especially if there are many terms. A good strategy is to draw arrows connecting the number outside the parentheses to each term inside, as a visual reminder to distribute to all terms.
Combining Unlike Terms
Lastly, be careful not to combine unlike terms. You can only add or subtract terms that have the same variable raised to the same power. For example, you can combine 14x and -12x because they both have 'x' to the power of 1, but you can't combine 14x with 8 because 8 is a constant term. Keep those variables and constants separate until the very end!
Practice Makes Perfect
So, there you have it! We've successfully found the equivalent expression of 2(7x + 4) - 4(3 + 3x - 2). Remember, the key to mastering algebra is practice, practice, practice. The more problems you solve, the more comfortable you'll become with these concepts. Try working through similar problems on your own, and don't be afraid to ask for help if you get stuck. You've got this! Keep up the great work, and I'll see you in the next math adventure!