Solving The Equation: A Step-by-step Guide

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Introduction

Hey guys! Today, we're going to dive into solving a fascinating equation. It might look a bit intimidating at first glance, but don't worry, we'll break it down step by step, making it super easy to understand. Our main goal here is to find the value of 'x' that makes this equation true. This involves simplifying both sides of the equation, combining like terms, and isolating 'x'. So, let's jump right in and make math a bit more fun!

Solving equations is a fundamental skill in algebra and mathematics, and mastering it opens doors to more advanced topics. The equation we're tackling today involves fractions, parentheses, and multiple terms, providing a great opportunity to enhance our problem-solving abilities. Understanding how to approach such equations not only improves our mathematical proficiency but also develops our logical thinking and analytical skills. So, stick with me, and let's unravel this mathematical puzzle together!

This equation is not just an abstract problem; it represents a real-world scenario where balancing both sides is crucial. Think of it as a balancing scale where the left side must always equal the right side. Our mission is to maintain this balance while simplifying and solving for 'x'. By understanding this concept, we can apply it to various fields, from engineering to economics, where equations are used to model and solve complex problems. So, let's get started and see how we can bring order to this mathematical expression!

The Equation

Let's start by clearly stating the equation we're going to solve:

23[2(x+1)−x+12]=5(x2−2x−16)\frac{2}{3}\left[2(x+1)-\frac{x+1}{2}\right]=5\left(\frac{x}{2}-\frac{2 x-1}{6}\right)

This equation involves fractions, parentheses, and the variable 'x', making it a classic example of an algebraic equation. Our task is to isolate 'x' on one side of the equation to find its value. To do this, we'll need to perform several algebraic operations, ensuring we maintain the balance of the equation at each step. So, let's begin our journey of simplification and solution!

Before we dive into the step-by-step solution, let's take a moment to appreciate the structure of the equation. We have terms inside brackets, fractions to deal with, and the variable 'x' appearing in multiple places. This complexity is what makes it an interesting challenge. By tackling this equation, we're not just solving for 'x'; we're also honing our skills in algebraic manipulation and critical thinking. Each step we take will bring us closer to the final answer, and with a bit of patience and careful execution, we'll conquer this equation.

Remember, every equation tells a story, and this one is no different. It's a story of balance and equality, where both sides are interconnected and must remain in harmony. As we progress through the solution, we'll uncover the narrative hidden within the symbols and numbers. Math is not just about formulas and calculations; it's about understanding relationships and patterns. So, let's embark on this mathematical adventure and discover the solution to this intriguing equation!

Step 1: Simplify Inside the Parentheses

First, let's simplify the expressions inside the parentheses on both sides of the equation. On the left side, we have 2(x+1)2(x+1) and x+12\frac{x+1}{2}. Let's distribute the 2 in the first term:

2(x+1)=2x+22(x+1) = 2x + 2

So, the left side of the equation now looks like this:

23[2x+2−x+12]\frac{2}{3}\left[2x + 2 - \frac{x+1}{2}\right]

Now, let's work on the right side of the equation. We have x2−2x−16\frac{x}{2} - \frac{2x-1}{6}. To combine these fractions, we need a common denominator, which is 6. So, let's rewrite x2\frac{x}{2} with a denominator of 6:

x2=3x6\frac{x}{2} = \frac{3x}{6}

Now, the right side of the equation looks like this:

5(3x6−2x−16)5\left(\frac{3x}{6} - \frac{2x-1}{6}\right)

Simplifying inside the parentheses is crucial because it reduces the complexity of the equation and makes subsequent steps easier. By distributing and finding common denominators, we're essentially organizing the terms and preparing them for further manipulation. This step is like setting the stage for the main act, ensuring that all the elements are in place for a smooth performance. So, with the parentheses simplified, we're one step closer to solving for 'x'.

Remember, in mathematics, order of operations is key. Parentheses always come first, followed by exponents, multiplication and division, and finally, addition and subtraction. By adhering to this order, we ensure accuracy and consistency in our calculations. This meticulous approach is what separates a correct solution from a wrong one. So, let's continue to follow this principle as we move forward, simplifying and solving with precision.

As we simplified the expressions inside the parentheses, we not only made the equation more manageable but also gained a clearer view of the terms involved. This is often the case in problem-solving; breaking down a complex problem into smaller, simpler parts can make the overall solution more accessible. Each simplification is a small victory, and these victories add up to the final triumph of solving the equation. So, let's celebrate this progress and move on to the next step with confidence.

Step 2: Combine Fractions

Now, let's combine the fractions within the brackets on both sides. On the left side, we have:

2x+2−x+122x + 2 - \frac{x+1}{2}

To combine these terms, we need a common denominator, which is 2. So, let's rewrite 2x+22x + 2 with a denominator of 2:

2x+2=2(2x+2)2=4x+422x + 2 = \frac{2(2x + 2)}{2} = \frac{4x + 4}{2}

Now, we can combine the fractions:

4x+42−x+12=(4x+4)−(x+1)2=4x+4−x−12=3x+32\frac{4x + 4}{2} - \frac{x+1}{2} = \frac{(4x + 4) - (x + 1)}{2} = \frac{4x + 4 - x - 1}{2} = \frac{3x + 3}{2}

So, the left side of the equation now looks like this:

23[3x+32]\frac{2}{3}\left[\frac{3x + 3}{2}\right]

On the right side, we have:

3x6−2x−16\frac{3x}{6} - \frac{2x-1}{6}

Since they already have a common denominator, we can combine the fractions:

3x−(2x−1)6=3x−2x+16=x+16\frac{3x - (2x - 1)}{6} = \frac{3x - 2x + 1}{6} = \frac{x + 1}{6}

So, the right side of the equation now looks like this:

5(x+16)5\left(\frac{x + 1}{6}\right)

Combining fractions is a key step in simplifying equations that involve fractional terms. By finding a common denominator and combining numerators, we consolidate multiple fractions into a single, more manageable term. This process not only reduces the visual complexity of the equation but also streamlines the subsequent algebraic manipulations. It's like tidying up a messy room, making it easier to navigate and find what we need. So, with the fractions combined, we're making significant progress towards our goal.

The process of finding a common denominator and combining fractions might seem like a minor detail, but it's actually a fundamental skill in mathematics. It reinforces our understanding of fractional arithmetic and prepares us for more complex operations. Mastering these basic skills is essential for building a strong foundation in algebra and beyond. So, let's continue to pay attention to these details as we move forward, ensuring accuracy and efficiency in our calculations.

As we combined the fractions, we transformed the equation into a more concise and coherent form. This transformation is a testament to the power of mathematical operations to simplify and clarify complex expressions. Math is not just about numbers; it's about patterns and relationships, and by combining like terms, we're revealing these patterns and relationships more clearly. So, let's appreciate the elegance of this simplification and move on to the next step with renewed enthusiasm.

Step 3: Multiply and Simplify

Now, let's multiply the fractions and simplify both sides of the equation. On the left side, we have:

23[3x+32]\frac{2}{3}\left[\frac{3x + 3}{2}\right]

Multiply the fractions:

2(3x+3)3(2)=6x+66\frac{2(3x + 3)}{3(2)} = \frac{6x + 6}{6}

Now, simplify by dividing both the numerator and the denominator by 6:

6x+66=x+1\frac{6x + 6}{6} = x + 1

So, the left side of the equation is now:

x+1x + 1

On the right side, we have:

5(x+16)5\left(\frac{x + 1}{6}\right)

Multiply the fraction:

5(x+1)6=5x+56\frac{5(x + 1)}{6} = \frac{5x + 5}{6}

So, the right side of the equation is now:

5x+56\frac{5x + 5}{6}

Now, our equation looks much simpler:

x+1=5x+56x + 1 = \frac{5x + 5}{6}

Multiplying and simplifying is a crucial step in unraveling the equation. By performing these operations, we're essentially clearing out the clutter and bringing the core terms into sharper focus. This process allows us to see the underlying structure of the equation more clearly and strategize the next steps more effectively. It's like decluttering a workspace, creating a more conducive environment for problem-solving.

Simplification is a recurring theme in mathematics, and for good reason. It's the art of making complex things simple, of reducing problems to their essence. By simplifying, we not only make calculations easier but also gain deeper insights into the mathematical relationships at play. This skill is invaluable not just in math but in various aspects of life, where simplifying complex situations can lead to clearer understanding and better decision-making. So, let's embrace the power of simplification as we continue our journey.

As we multiplied and simplified, we witnessed the equation transforming into a more elegant and manageable form. This transformation is a testament to the beauty of mathematical operations to distill and clarify. Math is often seen as a language, and in this language, simplification is akin to refining our vocabulary, choosing the most precise and impactful words to convey our message. So, let's celebrate this progress and move on to the next step with a renewed appreciation for the art of simplification.

Step 4: Clear the Fraction

To get rid of the fraction on the right side, we can multiply both sides of the equation by 6:

6(x+1)=6(5x+56)6(x + 1) = 6\left(\frac{5x + 5}{6}\right)

This gives us:

6x+6=5x+56x + 6 = 5x + 5

Now, we have an equation without fractions, which is much easier to solve.

Clearing the fraction is a strategic move that simplifies the equation significantly. Fractions can often be a source of complexity and confusion, so eliminating them early in the process can streamline the solution. This step is like removing an obstacle from our path, allowing us to move forward more smoothly and confidently. By multiplying both sides of the equation by the denominator, we maintain the balance while transforming the equation into a more manageable form.

This step highlights the importance of maintaining balance in equations. Whatever operation we perform on one side, we must also perform on the other side to preserve the equality. This principle of balance is fundamental to algebra and serves as a guiding light in our problem-solving journey. It's like a seesaw; if we add weight to one side, we must add the same weight to the other side to keep it level. So, let's always keep this balance in mind as we manipulate equations.

As we cleared the fraction, we witnessed the equation taking on a simpler and more familiar form. This transformation is a testament to the power of algebraic manipulation to make complex problems more accessible. Math is often seen as a puzzle, and in this puzzle, clearing the fraction is like finding a key piece that unlocks the next stage of the solution. So, let's celebrate this breakthrough and move on to the next step with a renewed sense of purpose.

Step 5: Isolate x

Now, let's isolate 'x' on one side of the equation. Subtract 5x from both sides:

6x+6−5x=5x+5−5x6x + 6 - 5x = 5x + 5 - 5x

This simplifies to:

x+6=5x + 6 = 5

Now, subtract 6 from both sides:

x+6−6=5−6x + 6 - 6 = 5 - 6

This gives us:

x=−1x = -1

So, the solution to the equation is x=−1x = -1.

Isolating 'x' is the ultimate goal of solving an equation. It's the moment when we finally uncover the value that satisfies the equation and makes it true. This step is like reaching the summit of a mountain after a challenging climb; it's the culmination of all our efforts and a moment of triumph. By carefully applying algebraic operations, we've managed to separate 'x' from all other terms and reveal its true value.

The process of isolating 'x' involves strategically moving terms across the equals sign, always maintaining the balance of the equation. This requires a keen understanding of inverse operations; for example, we use subtraction to undo addition and vice versa. Mastering these operations is essential for becoming proficient in algebra and problem-solving. It's like learning the rules of a game, allowing us to play with skill and confidence.

As we isolated 'x', we experienced the satisfaction of completing the puzzle and finding the solution. This feeling is a powerful motivator in mathematics and encourages us to tackle more challenging problems. Math is not just about finding answers; it's about the journey of discovery and the sense of accomplishment that comes with it. So, let's celebrate this moment of victory and reflect on the steps we took to reach this point.

Conclusion

We have successfully solved the equation 23[2(x+1)−x+12]=5(x2−2x−16)\frac{2}{3}\left[2(x+1)-\frac{x+1}{2}\right]=5\left(\frac{x}{2}-\frac{2 x-1}{6}\right) and found that x=−1x = -1. We did this by simplifying the expressions inside the parentheses, combining fractions, multiplying and simplifying, clearing the fraction, and finally, isolating 'x'.

Solving equations is a fundamental skill in mathematics and a valuable tool for problem-solving in various fields. By breaking down the equation into manageable steps and applying algebraic principles, we were able to find the solution. Remember, practice makes perfect, so keep solving equations to sharpen your skills!

Congratulations, guys! You've made it to the end of this mathematical journey. We've not only solved an equation but also reinforced our understanding of algebraic principles and problem-solving strategies. This knowledge is a valuable asset that will serve you well in future mathematical endeavors and beyond. So, let's continue to explore the fascinating world of mathematics with curiosity and confidence.

In this article, we've walked through the step-by-step process of solving a complex algebraic equation. We've seen how simplification, strategic manipulation, and a clear understanding of algebraic principles can lead us to the solution. Math is not just a collection of formulas and rules; it's a way of thinking, a way of approaching problems with logic and precision. So, let's carry this mindset forward and continue to conquer mathematical challenges with enthusiasm and determination.