Equivalent Expressions For -2(5x - 3/4) A Step-by-Step Guide

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Hey everyone! Let's break down this math problem together. We're trying to figure out which expressions are the same as −2(5x−34)-2(5x - \frac{3}{4}). This involves using the distributive property and simplifying. Don't worry, it's not as scary as it sounds! We'll go step by step to make sure we get it right. So, let's put on our thinking caps and dive into the world of equivalent expressions!

Decoding the Distributive Property

At the heart of this problem is the distributive property. This mathematical principle allows us to multiply a single term by multiple terms inside parentheses. It states that a(b + c) = ab + ac. Simply put, we multiply the term outside the parentheses by each term inside the parentheses individually. This is crucial for understanding how to expand and simplify expressions like the one we have. In our case, we need to distribute the -2 across both the 5x and the -3/4 inside the parentheses. This means we'll be multiplying -2 by both terms, which will lead us to an equivalent, expanded form of the expression. Mastering the distributive property opens up a whole new world of algebraic manipulations, making complex equations more manageable and understandable. So, let's gear up and apply this property to unravel our original expression!

Understanding the distributive property is the key to unlocking this problem. Remember, it's all about multiplying the term outside the parentheses by each term inside. In the expression −2(5x−34)-2(5x - \frac{3}{4}), we need to distribute the -2 to both the 5x and the -\frac{3}{4}. This means we'll have two multiplications to perform: (-2) * (5x) and (-2) * (-3/4). The result of these multiplications will form the equivalent expression, which we can then compare with the options provided. So, let's move forward and perform these multiplications to see what we get. This step-by-step approach will make the problem much clearer and easier to solve.

The distributive property might seem like a simple concept, but it's a fundamental tool in algebra. It's used extensively in solving equations, simplifying expressions, and even in more advanced mathematical concepts. Think of it as a mathematical Swiss Army knife – versatile and essential! The beauty of this property lies in its ability to transform a seemingly complex expression into a more manageable form. By distributing the term outside the parentheses, we effectively break down the expression into smaller, easier-to-handle pieces. This makes simplification much more straightforward. So, remember, when you see an expression with parentheses and a term multiplying it, your first instinct should be to apply the distributive property. It's your secret weapon for tackling algebraic challenges!

Applying the Distributive Property to -2(5x - 3/4)

Okay, let's get our hands dirty and apply the distributive property to our expression: −2(5x−34)-2(5x - \frac{3}{4}). As we discussed, this means multiplying -2 by both 5x and -\frac{3}{4}. Let's start with the first multiplication: (-2) * (5x). This is straightforward – we multiply the coefficients, -2 and 5, to get -10. So, the first term becomes -10x. Now, let's tackle the second multiplication: (-2) * (-3/4). Here, we're multiplying two negative numbers, which results in a positive number. To multiply a whole number by a fraction, we can think of the whole number as a fraction with a denominator of 1. So, -2 can be written as -2/1. Multiplying -2/1 by -3/4 gives us (2 * 3) / (1 * 4) = 6/4. So, the second term becomes +6/4. Putting it all together, our expanded expression is -10x + 6/4. This is a crucial step, and now we can compare this result with the options provided.

Now that we've applied the distributive property, our expression looks like this: −10x+64-10x + \frac{6}{4}. It's important to remember that fractions can often be simplified. In this case, \frac{6}{4} can be reduced to a simpler form. Both the numerator (6) and the denominator (4) are divisible by 2. Dividing both by 2, we get \frac{6 \div 2}{4 \div 2} = \frac{3}{2}. So, our simplified expression becomes −10x+32-10x + \frac{3}{2}. This simplification step is essential because some of the answer choices might be in the simplified form, and we want to make sure we're comparing apples to apples. By simplifying the fraction, we make it easier to identify the equivalent expressions among the options provided. So, keep in mind that simplifying fractions is often a necessary step in these kinds of problems.

Applying the distributive property is like opening a treasure chest – it reveals the hidden form of the expression. In our case, we transformed −2(5x−34)-2(5x - \frac{3}{4}) into −10x+64-10x + \frac{6}{4}, and further simplified it to −10x+32-10x + \frac{3}{2}. This process not only helps us solve the problem at hand but also reinforces our understanding of algebraic manipulation. Remember, mathematics is like a language – the more you practice, the more fluent you become. So, embrace the process of applying these properties and simplifying expressions. It's a journey of discovery, and each step you take strengthens your mathematical muscles!

Evaluating the Given Expressions

Let's take a look at the expressions provided and see which ones match our simplified expression, −10x+32-10x + \frac{3}{2}.

  1. −2(5x)+(−2)(−34)-2(5x) + (-2)(-\frac{3}{4}): This expression represents the initial step of applying the distributive property, before any simplification. Let's break it down. (-2) * (5x) equals -10x. And (-2) * (-3/4) equals +6/4, which simplifies to +3/2. So, this expression is equivalent to -10x + 3/2. This one matches our simplified expression!

  2. −10x−34-10x - \frac{3}{4}: This expression has the correct x-term (-10x), but the constant term is -3/4, which is different from our +3/2. So, this one is not equivalent.

  3. −10x+62-10x + \frac{6}{2}: This expression has the correct x-term (-10x). Let's simplify the fraction 6/2. 6 divided by 2 is 3. So, this expression is equivalent to -10x + 3. This is not the same as -10x + 3/2. So, this one is not equivalent either.

  4. −10x+32-10x + \frac{3}{2}: This expression has the correct x-term (-10x) and the correct constant term (+3/2). This one matches our simplified expression!

  5. −10x−62-10x - \frac{6}{2}: This expression has the correct x-term (-10x), but the constant term is -6/2, which simplifies to -3. This is different from our +3/2. So, this one is not equivalent.

It's like a mathematical treasure hunt, where we're searching for the expressions that perfectly match our simplified result. By carefully evaluating each option, we can identify the ones that are truly equivalent to our original expression.

Final Answer: The Equivalent Expressions

Alright, guys, we've done the hard work! We applied the distributive property, simplified the expression, and carefully evaluated each option. Based on our analysis, the expressions that are equivalent to −2(5x−34)-2(5x - \frac{3}{4}) are:

  • −2(5x)+(−2)(−34)-2(5x) + (-2)(-\frac{3}{4})
  • −10x+32-10x + \frac{3}{2}

These are the winners! By going through this process, we've not only solved the problem but also strengthened our understanding of equivalent expressions and the distributive property. Remember, math is a journey, not a destination. Keep practicing, keep exploring, and you'll become a mathematical wizard in no time!

So, there you have it! We've successfully navigated the world of equivalent expressions and emerged victorious. Give yourselves a pat on the back – you've earned it! Remember, the key to success in math is understanding the fundamental principles and practicing consistently. So, keep those mathematical gears turning, and you'll be amazed at what you can achieve!