Solving For X In Equations A Comprehensive Guide
In the realm of mathematics, solving for x in equations is a fundamental skill. It's the cornerstone of algebra and crucial for tackling more complex mathematical problems. This article delves into the intricacies of solving for x in various equations, providing a comprehensive guide with step-by-step explanations and examples. Whether you're a student grappling with algebraic concepts or simply seeking to refresh your math skills, this guide will equip you with the knowledge and techniques to confidently solve for x.
Let's embark on this mathematical journey by unraveling the process of solving for x, empowering you to conquer equations with ease.
1. Unveiling the Equation x × (5 - 3/x) - 4 × (3 - x) = ?
To effectively solve for x in the equation x × (5 - 3/x) - 4 × (3 - x) = ?, we must embark on a meticulous journey through the realms of algebraic manipulation. This equation, at first glance, may appear daunting, but with a systematic approach, we can unravel its complexities and arrive at a solution. Our initial step involves distributing the x and the -4 across the parentheses, setting the stage for simplification and the eventual isolation of x. This distribution is not merely a mechanical process; it's a strategic maneuver that allows us to transform the equation into a more manageable form, paving the way for the application of further algebraic techniques. As we delve deeper into the equation, we'll encounter opportunities to combine like terms, further streamlining the expression and bringing us closer to our goal of solving for x. Each step we take is a deliberate stride towards clarity, a testament to the power of methodical problem-solving in mathematics. By adhering to the fundamental principles of algebra and employing careful attention to detail, we can navigate the equation's intricacies and emerge with a triumphant solution.
Initial Transformation:
Our first step in unraveling this equation is to distribute the terms outside the parentheses. This involves multiplying x by both 5 and -3/x, and multiplying -4 by both 3 and -x. This seemingly simple act of distribution is a pivotal step, as it allows us to eliminate the parentheses and bring all the terms into a single, unified expression. It's like unwrapping a complex package, revealing the individual components that make up the whole. By carefully executing this distribution, we set the stage for further simplification and manipulation, bringing us closer to our ultimate goal of solving for x.
x × 5 - x × (3/x) - 4 × 3 + 4 × x = 0
This simplifies to:
5x - 3 - 12 + 4x = 0
Combining Like Terms:
Now that we've successfully distributed the terms and eliminated the parentheses, we can take the next step towards simplification: combining like terms. This involves identifying terms that share the same variable or are constants, and then adding or subtracting them as appropriate. In our equation, we have two terms with x (5x and 4x) and two constant terms (-3 and -12). By combining these like terms, we effectively consolidate the equation, making it more concise and easier to manipulate. This process is akin to organizing a cluttered workspace, grouping similar items together to create a sense of order and clarity. The result is a streamlined equation that brings us closer to isolating x and finding its value.
Combining the x terms (5x + 4x) and the constant terms (-3 - 12), we get:
9x - 15 = 0
Isolating x:
With the equation simplified to 9x - 15 = 0, the next crucial step is to isolate x. This means manipulating the equation to get x by itself on one side. To achieve this, we'll employ the fundamental principle of maintaining balance: whatever operation we perform on one side of the equation, we must also perform on the other. Our strategy here involves first adding 15 to both sides, effectively canceling out the -15 on the left side. This isolates the term with x, bringing us closer to our goal. Then, we'll divide both sides by 9, the coefficient of x, to finally isolate x completely. This process is akin to peeling away layers to reveal the core essence, in this case, the value of x. Each step is a deliberate move, guided by the principles of algebraic manipulation, leading us to the solution we seek.
To isolate x, we add 15 to both sides:
9x = 15
Then, we divide both sides by 9:
x = 15/9
Simplifying the Solution:
We've successfully isolated x and found its value to be 15/9. However, in mathematics, we always strive for the most concise and elegant representation of our solutions. This often means simplifying fractions to their lowest terms. In this case, both the numerator (15) and the denominator (9) share a common factor: 3. By dividing both by 3, we can reduce the fraction to its simplest form. This act of simplification is not merely cosmetic; it reveals the underlying mathematical truth in its purest form. It's like polishing a gemstone to reveal its brilliance, showcasing the inherent beauty of the solution.
x = 15/9 can be simplified by dividing both the numerator and denominator by 3:
x = 5/3
Therefore, the solution to the equation x × (5 - 3/x) - 4 × (3 - x) = ? is:
Answer: D) 5/3
2. Solving the Equation (3/7) - 4 × (x/3 - 1/7) = 2 × (5x/6 + 3)
Now, let's tackle the second equation: (3/7) - 4 × (x/3 - 1/7) = 2 × (5*x/6 + 3). This equation presents a slightly more intricate challenge with fractions and parentheses, but the fundamental principles of algebra remain our steadfast guide. Our initial strategy involves distributing the terms outside the parentheses, a crucial step in unraveling the equation's complexity. This distribution allows us to eliminate the parentheses and consolidate the terms into a more manageable form. It's like clearing away the underbrush in a forest, revealing the underlying structure of the trees. As we distribute, we must pay careful attention to the signs and coefficients, ensuring accuracy in our calculations. This meticulous approach sets the stage for further simplification, as we prepare to combine like terms and ultimately isolate x. Each step we take is a deliberate stride towards clarity, a testament to the power of methodical problem-solving in mathematics. By adhering to the fundamental principles of algebra and employing careful attention to detail, we can navigate the equation's intricacies and emerge with a triumphant solution.
Initial Transformation
As with the previous equation, our journey to solve for x begins with distributing the terms outside the parentheses. This crucial step allows us to dismantle the equation's initial complexity and pave the way for simplification. In this case, we'll distribute the -4 across the terms inside the first set of parentheses and the 2 across the terms inside the second set. This process is akin to carefully disassembling a machine, separating its components to gain a better understanding of how they interact. As we distribute, we must be mindful of the signs and coefficients, ensuring accuracy in our calculations. This meticulous approach lays the foundation for the subsequent steps, as we prepare to combine like terms and ultimately isolate x.
(3/7) - 4 × (x/3) + 4 × (1/7) = 2 × (5*x/6) + 2 × 3
This simplifies to:
(3/7) - (4x/3) + (4/7) = (5x/3) + 6
Combining Like Terms (Constants):
Now that we've successfully distributed the terms and eliminated the parentheses, our next step is to consolidate the constant terms on the left side of the equation. This involves identifying the terms that do not contain the variable x and combining them. In this case, we have two constant terms: 3/7 and 4/7. By adding these fractions, we simplify the equation, making it more manageable and bringing us closer to our goal of isolating x. This process is akin to organizing a cluttered desk, grouping similar items together to create a sense of order and clarity. The result is a streamlined equation that allows us to focus on the remaining terms and strategize our next steps in solving for x.
Combining the constants (3/7) + (4/7) gives us:
(7/7) - (4x/3) = (5x/3) + 6
Which further simplifies to:
1 - (4x/3) = (5x/3) + 6
Moving x Terms to One Side:
With the constant terms consolidated, our next strategic move is to gather all the terms containing x onto one side of the equation. This is a crucial step in isolating x, as it allows us to treat all the x terms as a single entity. To achieve this, we'll add (4x/3) to both sides of the equation, effectively canceling out the -(4x/3) term on the left side. This process is akin to herding scattered sheep into a single flock, bringing them together for easier management. By collecting the x terms on one side, we create a more focused equation, setting the stage for the final steps in solving for x.
To move the x terms, we add (4*x/3) to both sides:
1 = (5x/3) + (4x/3) + 6
This simplifies to:
1 = (9*x/3) + 6
Further simplifying the fraction, we get:
1 = 3x + 6
Isolating x:
Now that we have all the x terms on one side and the constants on the other, we can proceed to isolate x completely. This involves a two-step process: first, we'll subtract 6 from both sides to isolate the term with x. Then, we'll divide both sides by 3, the coefficient of x, to finally obtain the value of x. This process is akin to carefully dismantling a complex structure, removing each component until only the core remains. Each step is a deliberate move, guided by the principles of algebraic manipulation, leading us to the ultimate solution.
To isolate x, we subtract 6 from both sides:
1 - 6 = 3x
-5 = 3x
Then, we divide both sides by 3:
x = -5/3
Therefore, the solution to the equation (3/7) - 4 × (x/3 - 1/7) = 2 × (5x/6 + 3) is:
Answer: A) -5/3
Conclusion
In this comprehensive guide, we've meticulously dissected two equations, demonstrating the step-by-step process of solving for x. We've explored the fundamental techniques of distribution, combining like terms, and isolating x, emphasizing the importance of a systematic and methodical approach. Each step we've taken has been a deliberate stride towards clarity, guided by the principles of algebraic manipulation. By mastering these techniques, you'll be well-equipped to tackle a wide range of equations and confidently solve for x in various mathematical contexts. Remember, practice is key to solidifying your understanding and honing your problem-solving skills. So, embrace the challenge, delve into the world of equations, and unlock your mathematical potential.
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