Simplifying Expressions: Finding Equivalents To -2(6r)

Hey math enthusiasts! Today, we're diving into the world of algebraic expressions and figuring out which ones are equivalent to $-2(6r)$. Don't worry, it's not as scary as it sounds! We'll break it down step-by-step, making sure everyone understands the concepts. So, grab your pencils, and let's get started. Understanding the concept of equivalent expressions is crucial in algebra. Basically, equivalent expressions look different but have the same value, no matter what number we plug in for our variable, in this case, 'r'. Think of it like different ways to write the same thing. For example, 2 + 2 is the same as 4. They're just different representations of the same value. In algebra, we often use operations like distribution, combining like terms, and factoring to transform expressions into equivalent forms. Let's explore each option provided to determine which ones are equivalent to the original expression, $-2(6r)$. Remember that the core of simplifying expressions lies in applying the correct rules of arithmetic and algebra. We're going to examine each option carefully, ensuring we follow these rules to derive accurate results. The main goal here is to identify expressions that produce the same numerical value as our original expression for any given value of 'r'. This process helps us build a strong foundation in algebraic manipulation, which is essential for more advanced math concepts. Ready to roll up our sleeves and solve these expressions? Let's get started. Let's make sure we go through all the steps to ensure that the answer we choose is the correct one, and that we understand the process clearly. Ready to see the answer, guys?

Option A: Understanding and Simplifying -12r

Alright, let's start with option A: $-12r$. This one is pretty straightforward. To see if it's equivalent to $-2(6r)$, we need to simplify the original expression first. The original expression, $-2(6r)$, involves multiplication. When you have a number directly outside parentheses next to terms inside the parentheses, it means we need to multiply the number outside by each term inside. This is called the distributive property. In our case, we'll multiply $-2$ by $6r$. So, $-2 imes 6r = -12r$. See that, guys? We're taking the expression and working to rewrite it! Pretty easy, right? Comparing this simplified form to option A, we see that $-2(6r)$ simplifies directly to $-12r$. Therefore, option A, which is also $-12r$, is equivalent to the original expression. Remember, equivalent expressions produce the same result for any value of the variable. Let's take a quick example. If $r = 1$, then $-2(6 imes 1) = -2(6) = -12$. Also, $-12 imes 1 = -12$. If $r = 2$, then $-2(6 imes 2) = -2(12) = -24$. And $-12 imes 2 = -24$. The result is the same! Option A checks out. This shows us how the distributive property works to simplify expressions and make them easier to work with. In this case, we've shown that the distributive property simplifies the expression to an equivalent form, making it easy to identify the correct answer among a set of options. Make sure to remember that, it is very important.

Option B: Evaluating r - 12

Now, let's move on to option B: $r - 12$. This one is different from the original expression, $-2(6r)$, which we already know simplifies to $-12r$. Option B, $r - 12$, is a subtraction problem. To see if this expression is equivalent, we can try substituting some values for 'r' and comparing the results. When $r = 1$, the original expression, $-12r$, becomes $-12(1) = -12$. If we use the same value in option B, $r - 12$, we get $1 - 12 = -11$. Notice that when we substitute $r = 1$, we get a different answer when we use the original expression vs. when we use Option B. Because the results are different, we can see that $r - 12$ is not equivalent to $-12r$. This tells us that option B is not a correct answer. It is very important to carefully substitute values and compare the results when you are trying to find equivalent values. Remember, to be equivalent, the expressions must give the same result for all possible values of the variable. Let's try another example. Let's plug in $r = 0$. Using the original expression, we get $-12(0) = 0$. Plugging this into option B, we get $0 - 12 = -12$. The results are different, so $r - 12$ cannot be equivalent to $-12r$. This process highlights why careful, step-by-step simplification and evaluation are so crucial in algebra. Let's not make careless mistakes, guys! We're doing great!

Option C: Analyzing 6r - 2

Time to examine option C: $6r - 2$. This expression has both a variable term ($6r$) and a constant term ($-2$). To determine if it's equivalent to our simplified original expression, $-12r$, we can use the same strategy as before: substituting values for 'r'. If we let $r = 1$, then $-12r$ becomes $-12(1) = -12$. Now, let's plug $r = 1$ into option C: $6(1) - 2 = 6 - 2 = 4$. So, when $r = 1$, the original expression evaluates to $-12$, but option C evaluates to 4. Since the results are different, we can conclude that option C is not equivalent to the original expression. Let's try another value, say $r = 2$. For the original expression, we have $-12(2) = -24$. For option C, we have $6(2) - 2 = 12 - 2 = 10$. The different results for each value of 'r' confirm that option C is not equivalent. Let's think about why this is the case. The original expression, $-2(6r)$, involves multiplication. Option C, $6r - 2$, involves multiplication, but also involves subtraction. Different operations result in different values, so it's critical to ensure that we apply the rules in the correct order to find the accurate solution. Understanding these operational differences is key to mastering algebra. Are you still with me, guys? We are almost there!

Option D: Deconstructing (6 + r) * -2

Finally, let's tackle option D: $(6 + r) imes -2$. This expression uses the distributive property as well, but in a different way. To see if it's equivalent to $-12r$, which is what our original expression, $-2(6r)$, simplifies to, we need to simplify this expression. Applying the distributive property, we multiply $-2$ by both terms inside the parentheses: $(6 imes -2) + (r imes -2)$. This simplifies to $-12 - 2r$. Let's compare this with $-12r$. The expressions are not identical. Option D, $-12 - 2r$, includes both a constant term, $-12$, and a variable term, $-2r$. Remember, our goal is to find equivalent expressions. Therefore, because of the constant term of -12, option D is not equivalent to $-12r$. Let's choose the same strategy to make sure. Let $r = 1$. The original expression simplifies to $-12(1) = -12$. Option D becomes $(6 + 1) imes -2 = 7 imes -2 = -14$. They are not the same! Let's choose another number. If $r = 0$, then the original expression becomes $-12(0) = 0$. Option D is $(6 + 0) imes -2 = 6 imes -2 = -12$. This confirms our understanding: option D is not equivalent to the original expression. This step reinforces the idea of how important it is to be careful when applying the rules of algebra. Sometimes it can be easy to make a mistake when following the rules. So let's all make sure that we take our time and carefully check our answers. You can do it!

Conclusion: Identifying the Equivalent Expression

Alright, we've carefully evaluated each option. After simplifying the original expression, $-2(6r)$, we found it simplifies to $-12r$. Option A, which is $-12r$, is the only expression that is equivalent to the original expression. Options B, C, and D were not equivalent because they did not simplify to the same value for every value of 'r'. This exercise highlights the importance of understanding and applying the distributive property, simplifying expressions, and evaluating expressions correctly. Remember, guys, practice is key! The more you work with these concepts, the easier they will become. Keep up the awesome work, and keep exploring the amazing world of mathematics!