Solving Equivalent Expressions In Equations A Step-by-Step Guide

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Hey there, math enthusiasts! Today, we're diving into an exciting algebraic equation that might seem a bit tangled at first glance, but trust me, we'll untangle it together. The challenge is to find an equivalent expression for the equation $4(3x-6) + 2(3x-1) = 2$. To do this, we'll meticulously walk through the steps of simplifying the equation. Understanding the basics of algebraic manipulation is crucial in solving such problems, and by the end of this guide, you'll not only know the correct answer but also grasp the methodology behind it. So, let's roll up our sleeves and get started!

Breaking Down the Original Equation

The equation we're tackling is $4(3x-6) + 2(3x-1) = 2$. At first sight, it might look a bit daunting, but don't worry! We're going to break it down step by step. The key here is to remember our order of operations—PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Our first mission is to simplify the expressions within the parentheses by applying the distributive property. This means we'll multiply the numbers outside the parentheses by each term inside.

Step 1: Distribute the Constants

Let's start by distributing the 4 in the first term, $4(3x-6)$. This involves multiplying 4 by both $3x$ and $-6$. So, $4 * 3x$ equals $12x$, and $4 * -6$ equals $-24$. Therefore, the first term simplifies to $12x - 24$.

Now, let's move on to the second term, $2(3x-1)$. We'll distribute the 2 similarly, multiplying it by both $3x$ and $-1$. So, $2 * 3x$ equals $6x$, and $2 * -1$ equals $-2$. The second term, therefore, simplifies to $6x - 2$.

Step 2: Rewrite the Equation

After distributing the constants, our equation now looks like this: $12x - 24 + 6x - 2 = 2$. See? We've already made significant progress! The next step involves combining like terms to further simplify the equation. This means we'll group together the terms with 'x' and the constant terms separately.

Combining Like Terms for Simplification

Now that we've distributed and expanded our original equation, it's time to make things even simpler by combining like terms. This is a fundamental step in solving algebraic equations and helps in reducing the equation to its most basic form. Remember, like terms are those that have the same variable raised to the same power (in our case, terms with 'x') and the constant terms (numbers without variables).

Step 3: Grouping Like Terms

Looking at our equation, $12x - 24 + 6x - 2 = 2$, we can identify two types of terms: those with 'x' ($12x$ and $6x$) and constant terms ($-24$ and $-2$). Our goal is to group these together so we can combine them. It's like sorting apples and oranges—we want to put the 'x' terms with the 'x' terms and the constants with the constants.

So, let's rearrange our equation slightly to visually group these like terms: $(12x + 6x) + (-24 - 2) = 2$. Notice how we've just changed the order of the terms but haven't changed the equation itself. This is allowed because of the commutative property of addition, which states that the order in which numbers are added does not change the sum.

Step 4: Combining 'x' Terms

Now that our like terms are grouped, let's combine the 'x' terms first. We have $12x + 6x$. This is straightforward addition: 12 of something plus 6 of the same thing equals 18 of that thing. So, $12x + 6x$ simplifies to $18x$. We've successfully combined our 'x' terms into a single term!

Step 5: Combining Constant Terms

Next up, we'll combine our constant terms: $-24$ and $-2$. Remember, we're adding a negative number to another negative number. When we add two negative numbers, we move further into the negative realm. So, $-24 + (-2)$ is the same as $-24 - 2$, which equals $-26$. We've now combined our constant terms into a single constant.

Step 6: Rewrite the Simplified Equation

After combining like terms, our equation now looks much simpler. We've gone from a longer expression to something more manageable. Our equation $12x - 24 + 6x - 2 = 2$ has now been simplified to $18x - 26 = 2$. This is a significant step forward in finding the equivalent expression we're looking for.

Identifying the Equivalent Expression

We've done the hard work of simplifying the equation, and now comes the moment of truth: identifying the equivalent expression from the given options. This is where our step-by-step simplification pays off. We've transformed the original equation into a much simpler form, making it easier to compare with the provided choices.

Step 7: Compare Simplified Equation with Options

Our simplified equation is $18x - 26 = 2$. Now, let's take a look at the options given in the question:

A. $6x - 7 = 2$ B. $18x - 26 = 2$ C. $18x - 7 = 2$ D. $18x - 22 = 2$

By comparing our simplified equation with these options, we can clearly see which one matches exactly. It's like finding a matching puzzle piece—the shapes have to be identical.

Step 8: Select the Matching Option

Looking closely, we can see that option B, $18x - 26 = 2$, is an exact match to our simplified equation. The coefficients of 'x' are the same, the constant terms are the same, and the value on the right side of the equation is the same. This means that option B is the equivalent expression we've been searching for.

The other options don't match our simplified equation. Option A has a different coefficient for 'x' and a different constant term. Options C and D have the correct coefficient for 'x' but different constant terms. Therefore, they are not equivalent to the original equation.

Step 9: Confirm the Answer

To be absolutely sure, it's always a good idea to double-check our work. We started with $4(3x-6) + 2(3x-1) = 2$, distributed the constants, combined like terms, and arrived at $18x - 26 = 2$. Option B matches this simplified form perfectly, so we can confidently confirm that it is the correct answer.

Final Answer

After a thorough step-by-step simplification and comparison, we've successfully identified the equivalent expression for the given equation. The correct answer is:

B. $18x - 26 = 2$

We've journeyed through the process of distributing, combining like terms, and simplifying the equation. This wasn't just about finding the right answer; it was about understanding the methodology and the underlying principles of algebra. Algebraic manipulation is a fundamental skill in mathematics, and mastering these steps will help you tackle more complex problems with confidence.

Remember, the key to solving algebraic equations is to break them down into smaller, manageable steps. By carefully applying the rules of mathematics and paying attention to detail, you can unravel even the most intricate equations. Keep practicing, and you'll become a math whiz in no time!

Why is Understanding Equivalent Expressions Important?

Understanding equivalent expressions is more than just a mathematical exercise; it's a crucial skill that extends beyond the classroom. In this section, we'll delve into why this concept is so important and how it applies to real-world scenarios. Equivalent expressions are, at their core, different ways of writing the same mathematical idea. Think of them as different languages expressing the same thought. Being fluent in recognizing and manipulating these expressions can significantly enhance your problem-solving abilities and your understanding of mathematical concepts.

Simplification and Problem Solving

One of the primary reasons understanding equivalent expressions is important is for simplification. As we demonstrated in our original problem, complex equations can often be simplified into a more manageable form by using equivalent expressions. This simplification can make it much easier to solve for a variable, graph a function, or analyze a mathematical model. For instance, in fields like physics and engineering, complex formulas are often simplified using algebraic manipulation to make calculations more straightforward. Without the ability to recognize and create equivalent expressions, these simplifications would be much more challenging, if not impossible.

Conceptual Understanding

Working with equivalent expressions also deepens your conceptual understanding of mathematics. It's not just about memorizing formulas or procedures; it's about understanding why certain manipulations are valid and how different mathematical concepts are related. For example, understanding the distributive property and how it leads to equivalent expressions helps you see the underlying structure of algebraic equations. This deeper understanding allows you to approach problems with more flexibility and creativity, rather than relying on rote memorization.

Real-World Applications

The ability to work with equivalent expressions has numerous real-world applications. Consider scenarios in finance, where you might need to compare different loan options or investment strategies. Each option might be presented in a different form, but by using your knowledge of equivalent expressions, you can rewrite them in a standard format and make an accurate comparison. For example, you might need to compare a loan with a high interest rate and low fees to one with a low interest rate and high fees. By understanding equivalent expressions, you can calculate the total cost of each loan and make an informed decision.

Problem Solving in Everyday Life

Equivalent expressions also come into play in everyday problem-solving situations. For instance, when cooking, you might need to adjust a recipe to serve a different number of people. This often involves multiplying or dividing the quantities of ingredients, which is essentially creating equivalent expressions. Similarly, when planning a budget, you might need to calculate the cost of different options, such as buying in bulk versus buying individual items. By using your knowledge of equivalent expressions, you can make these calculations more efficiently and accurately.

Building a Foundation for Advanced Mathematics

Finally, understanding equivalent expressions is crucial for building a strong foundation for advanced mathematics. Many concepts in calculus, linear algebra, and differential equations rely heavily on the ability to manipulate algebraic expressions. For example, in calculus, you might need to simplify a complex expression before you can find its derivative or integral. In linear algebra, you might need to rewrite a matrix equation to solve for a vector. Without a solid understanding of equivalent expressions, these advanced topics can be much more difficult to grasp.

In conclusion, understanding equivalent expressions is a vital skill that extends far beyond the math classroom. It enhances your problem-solving abilities, deepens your conceptual understanding of mathematics, and has numerous real-world applications. Whether you're simplifying a complex equation, comparing financial options, or adjusting a recipe, the ability to recognize and manipulate equivalent expressions is a valuable asset. So, keep practicing, keep exploring, and keep building your mathematical toolkit!

Practice Problems to Sharpen Your Skills

Now that we've thoroughly covered the concept of equivalent expressions and their importance, it's time to put your knowledge to the test! Practice is key to mastering any mathematical skill, and working through additional problems will help solidify your understanding and build your confidence. In this section, we'll provide you with some practice problems that are similar to the one we tackled earlier. These problems will challenge you to apply the principles of distribution, combining like terms, and identifying equivalent expressions. So, grab a pencil and paper, and let's dive in!

Problem 1: Simplify and Find the Equivalent Expression

Question: Which of the following is equivalent to $3(2x + 4) - 2(x - 1) = 10$?

A. $4x + 10 = 10$ B. $8x + 14 = 10$ C. $4x + 14 = 10$ D. $8x + 10 = 10$

Hint: Remember to distribute the constants first, then combine like terms. Compare your simplified equation with the options provided to find the equivalent expression.

Problem 2: Solve for the Variable

Question: Find the value of x in the equation $5(x - 2) + 3(2x + 1) = 21$.

A. $x = 2$ B. $x = 3$ C. $x = 4$ D. $x = 5$

Hint: Simplify the equation by distributing and combining like terms. Then, isolate the variable x by performing the necessary algebraic operations.

Problem 3: Identify the Correct Simplification

Question: Which of the following is the correct simplification of the expression $2(4x - 3) - (x + 5)$?

A. $7x - 8$ B. $7x - 1$ C. $9x - 8$ D. $9x - 1$

Hint: Be careful with the negative sign in front of the parentheses. Remember to distribute it to both terms inside the parentheses.

Problem 4: Real-World Application

Question: A contractor is estimating the cost of tiling a rectangular floor. The floor is (3x + 2) feet long and (2x - 1) feet wide. The cost of the tile is $5 per square foot. Which expression represents the total cost of tiling the floor?

A. $30x^2 + 5x - 10$ B. $30x^2 + 5x - 5$ C. $30x^2 + 5x + 10$ D. $30x^2 + 10$

Hint: First, find the area of the rectangular floor by multiplying the length and width. Then, multiply the area by the cost per square foot.

Problem 5: Multi-Step Simplification

Question: Which of the following is equivalent to $4(2x - 1) + 2(3 - x) - 5x = 7$?

A. $x + 2 = 7$ B. $x - 2 = 7$ C. $x - 10 = 7$ D. $x + 10 = 7$

Hint: This problem involves multiple steps of distribution and combining like terms. Take your time and work through each step carefully.

Tips for Solving Practice Problems

• Read the question carefully: Make sure you understand what the question is asking before you start solving.
• Show your work: Write down each step of your solution. This will help you track your progress and identify any errors.
• Check your answer: Once you've found a solution, plug it back into the original equation to make sure it works.
• Practice regularly: The more you practice, the more comfortable you'll become with solving algebraic equations.
• Don't give up: If you're struggling with a problem, take a break and come back to it later. Sometimes a fresh perspective is all you need.

By working through these practice problems and following the tips above, you'll be well on your way to mastering the concept of equivalent expressions. Remember, practice makes perfect, so keep at it, and you'll become a confident problem solver in no time!

This comprehensive guide has walked you through the process of simplifying and identifying equivalent expressions in algebraic equations. From breaking down the original equation to comparing simplified forms with given options, each step has been explained in detail to ensure a clear understanding. We've also emphasized the importance of this skill in both mathematical and real-world contexts, and provided practice problems to help you hone your abilities. So, keep practicing, and you'll be simplifying equations like a pro in no time!