Simplify Algebraic Expressions: 5(1/3x + 7) - 3(1/2x - 4)

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Hey everyone, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of algebra to tackle an expression that might look a little intimidating at first glance: $5\left(\frac{1}{3} x+7\right)-3\left(\frac{1}{2} x-4\right)$. Don't worry, guys, we're going to break it down step-by-step, making it super easy to understand. Our main goal here is to find expressions that are equivalent to this original one. That means they'll have the exact same value no matter what number you plug in for 'x'. We'll be looking at a few options and figuring out which three are the real MVPs, the ones that are mathematically identical to our starting point. So, grab your notebooks, get ready to flex those brain muscles, and let's simplify this beast!

Understanding Equivalent Expressions

Alright, let's chat about what 'equivalent expressions' actually means in the land of math, especially when we're dealing with variables like 'x'. Basically, two expressions are equivalent if they produce the exact same output for any given input. Think of it like having two different recipes for the same delicious cake; they might use slightly different steps or ingredients listed in a different order, but in the end, you get the same amazing cake. In algebra, this means we can manipulate an expression, simplify it, or rewrite it in different forms, and as long as we follow the rules of mathematics, the new expression will be equivalent to the old one. The key tools we use for this are the distributive property, combining like terms, and order of operations. For our problem, we have the expression $5\left(\frac{1}{3} x+7\right)-3\left(\frac{1}{2} x-4\right)$. Our mission, should we choose to accept it, is to simplify this bad boy and then compare it to the given options to find the ones that match. This involves distributing the numbers outside the parentheses to the terms inside and then combining the 'x' terms and the constant terms. It's all about rearranging and simplifying without changing the fundamental value. So, when you see 'equivalent', just think 'same value, different look'. Let's get cracking on this!

Step-by-Step Simplification

Now, let's get down to business and simplify our original expression: $5\left(\frac{1}{3} x+7\right)-3\left(\frac{1}{2} x-4\right)$. The first thing we need to do is use the distributive property. This means multiplying the number outside each set of parentheses by each term inside those parentheses. So, for the first part, $5\left(\frac{1}{3} x+7\right)$, we multiply 5 by $\frac{1}{3} x$ and then multiply 5 by 7.

• $5 \times \frac{1}{3} x = \frac{5}{3} x$
• $5 \times 7 = 35$

So, the first part becomes $\frac{5}{3} x + 35$.

Now, let's tackle the second part: $-3\left(\frac{1}{2} x-4\right)$. It's crucial to include that minus sign when distributing the -3.

• $-3 \times \frac{1}{2} x = -\frac{3}{2} x$
• $-3 \times -4 = +12$

(Remember, a negative times a negative is a positive!).

So, the second part becomes $-\frac{3}{2} x + 12$.

Now, we combine the results from both parts:

$\frac{5}{3} x + 35 - \frac{3}{2} x + 12$

Our next move is to combine like terms. This means we group all the terms with 'x' together and all the constant terms (the numbers without 'x') together.

• Combine the 'x' terms: $\frac{5}{3} x - \frac{3}{2} x$
• Combine the constant terms: $35 + 12$

Let's work on the 'x' terms first. To subtract these fractions, we need a common denominator. The least common multiple of 3 and 2 is 6.

• $\frac{5}{3} x = \frac{5 \times 2}{3 \times 2} x = \frac{10}{6} x$
• $\frac{3}{2} x = \frac{3 \times 3}{2 \times 3} x = \frac{9}{6} x$

Now subtract: $\frac{10}{6} x - \frac{9}{6} x = \frac{10-9}{6} x = \frac{1}{6} x$

And for the constant terms, it's simple addition:

$35 + 12 = 47$

Putting it all together, our simplified expression is:

$\frac{1}{6} x + 47$

This is our target expression! Now we just need to compare this with the options provided to find the three equivalent ones. Get ready, because we're about to put our simplified version to the test against the choices. It's like a math showdown!

Evaluating the Options

Alright guys, we've done the heavy lifting and simplified our original expression to $\frac{1}{6} x + 47$. Now, the fun part begins: we need to go through each of the given options (A, B, C, and D) and see which ones simplify down to the same result. Remember, equivalent expressions always yield the same value. We'll break down each option, simplify it as much as possible, and then compare it to our benchmark $\frac{1}{6} x + 47$. Let's start with option A, then B, then C, and finally D.

Option A: $5 \frac{1}{3} x-3 \frac{1}{2} x+35-12$

First off, those mixed numbers look a bit funky in an algebraic expression, but we can convert them to improper fractions. Remember $5 \frac{1}{3}$ means $5 + \frac{1}{3}$ and $3 \frac{1}{2}$ means $3 + \frac{1}{2}$.

• $5 \frac{1}{3} = \frac{(5 \times 3) + 1}{3} = \frac{15+1}{3} = \frac{16}{3}$
• $3 \frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6+1}{2} = \frac{7}{2}$

So, option A becomes: $\frac{16}{3} x - \frac{7}{2} x + 35 - 12$.

Now, let's combine the 'x' terms: $\frac{16}{3} x - \frac{7}{2} x$. We need a common denominator, which is 6.

• $\frac{16}{3} x = \frac{16 \times 2}{3 \times 2} x = \frac{32}{6} x$
• $\frac{7}{2} x = \frac{7 \times 3}{2 \times 3} x = \frac{21}{6} x$

Subtracting them: $\frac{32}{6} x - \frac{21}{6} x = \frac{11}{6} x$.

Now, combine the constant terms: $35 - 12 = 23$.

So, option A simplifies to $\frac{11}{6} x + 23$. This is not equivalent to $\frac{1}{6} x + 47$. So, Option A is out, guys.

Option B: $\frac{1}{6} x+47$

Well, well, well, lookie here! Option B is exactly the same as our simplified expression. This means it is definitely equivalent to the original expression. So, Option B is a keeper! Bingo!

Option C: $1 \frac{2}{3} x+35-1 \frac{1}{2} x+12$

Let's convert these mixed numbers again. Remember, they are parts of the original expression, not necessarily the whole simplified form.

• $1 \frac{2}{3} = \frac{(1 \times 3) + 2}{3} = \frac{3+2}{3} = \frac{5}{3}$
• $1 \frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{2+1}{2} = \frac{3}{2}$

So, option C becomes: $\frac{5}{3} x + 35 - \frac{3}{2} x + 12$.

Let's combine the 'x' terms: $\frac{5}{3} x - \frac{3}{2} x$. We already did this when we simplified the original expression! The common denominator is 6.

• $\frac{5}{3} x = \frac{10}{6} x$
• $\frac{3}{2} x = \frac{9}{6} x$

Subtracting them: $\frac{10}{6} x - \frac{9}{6} x = \frac{1}{6} x$.

Now, combine the constant terms: $35 + 12 = 47$.

So, option C simplifies to $\frac{1}{6} x + 47$. This is also exactly the same as our simplified expression! Awesome! Option C is another keeper.

Option D: 5\left(\frac{1}{3} x\right)+5(7)-3\left(rac{1}{2} x\right)-3(-4)

This option looks like the intermediate step after applying the distributive property to both sets of parentheses in the original expression. Let's break it down:

• $5\left(\frac{1}{3} x\right) = \frac{5}{3} x$
• $5(7) = 35$
• $-3\left(\frac{1}{2} x\right) = -\frac{3}{2} x$
• $-3(-4) = +12$

So, option D becomes: $\frac{5}{3} x + 35 - \frac{3}{2} x + 12$.

Wait a second... this is the exact same expression we got right after distributing in our original simplification! And we know that simplifies to $\frac{1}{6} x + 47$. Therefore, Option D is also equivalent! Yes!

The Winning Trio

After carefully simplifying our original expression and evaluating each of the options, we've found our winners! The original expression $5\left(\frac{1}{3} x+7\right)-3\left(\frac{1}{2} x-4\right)$ simplifies to $\frac{1}{6} x + 47$. We found that the following options also simplify to the same form:

• Option B: $\frac{1}{6} x+47$ (This was identical to our simplified form).
• Option C: $1 \frac{2}{3} x+35-1 \frac{1}{2} x+12$ (Which we showed simplifies to $\frac{1}{6} x + 47$).
• Option D: 5\left(\frac{1}{3} x\right)+5(7)-3\left(rac{1}{2} x\right)-3(-4) (This represents the direct result of distributing the original terms, which also leads to $\frac{1}{6} x + 47$).

So, the three expressions equivalent to $5\left(\frac{1}{3} x+7\right)-3\left(\frac{1}{2} x-4\right)$ are B, C, and D. You guys absolutely crushed it! Understanding how to manipulate and simplify algebraic expressions is a super powerful skill, and you've just demonstrated that you've got it. Keep practicing, and you'll be an algebra whiz in no time. See you in the next one!