Solving The Equation 10x - 3(x - 6) = X + 30 A Step-by-Step Guide
Introduction: The Art of Equation Solving
In the realm of mathematics, equations serve as puzzles, each holding a secret value waiting to be discovered. The equation 10x - 3(x - 6) = x + 30 presents such a puzzle, inviting us to embark on a journey of algebraic manipulation to unveil the solution for x. This article delves into the step-by-step process of solving this equation, providing a clear and comprehensive guide for anyone seeking to master the art of equation solving. We'll not only dissect the mechanics but also explore the underlying principles that make this process both logical and elegant. Understanding these principles is key to tackling more complex equations and building a solid foundation in algebra.
The Initial Equation: A First Glance
Our starting point is the equation 10x - 3(x - 6) = x + 30. At first glance, it may seem daunting due to the presence of parentheses and multiple terms involving x. However, by systematically applying the rules of algebra, we can simplify this equation and isolate x. The first crucial step involves addressing the parentheses. The distributive property, a cornerstone of algebraic manipulation, allows us to multiply the -3 outside the parentheses by each term inside. This initial move is not just a mechanical step; it's a strategic maneuver that sets the stage for the rest of the solution. By removing the parentheses, we transform the equation into a more manageable form, paving the way for further simplification.
Applying the Distributive Property: Breaking Down the Barriers
The distributive property dictates that -3 multiplied by (x - 6) is equivalent to (-3 * x) + (-3 * -6). This yields -3x + 18. Substituting this back into the original equation, we get 10x - 3x + 18 = x + 30. This step is pivotal as it eliminates the parentheses, allowing us to combine like terms. The distributive property is not merely a trick; it's a fundamental principle rooted in the axioms of arithmetic. Understanding its application is crucial for simplifying expressions and equations across various mathematical contexts. It's a tool that transforms complex expressions into simpler, more manageable forms, making the subsequent steps of solving the equation significantly easier.
Combining Like Terms: Streamlining the Equation
Now, we focus on the left side of the equation: 10x - 3x + 18. Here, 10x and -3x are like terms, meaning they both contain the variable x raised to the same power (in this case, 1). We can combine these terms by simply adding their coefficients: 10 - 3 = 7. This gives us 7x + 18. Our equation now simplifies to 7x + 18 = x + 30. This step highlights the importance of recognizing and combining like terms, a skill that streamlines the equation and brings us closer to isolating x. Combining like terms is not just about simplifying; it's about revealing the underlying structure of the equation and making it easier to manipulate. This principle is applicable not just in algebra but across various branches of mathematics.
Isolating the Variable: The Quest for 'x'
Moving Terms: Balancing the Equation
Our goal is to isolate x on one side of the equation. To achieve this, we need to move all terms containing x to one side and all constant terms to the other. A common strategy is to subtract x from both sides of the equation: 7x + 18 - x = x + 30 - x. This maintains the equality because we're performing the same operation on both sides. This step is crucial because it begins the process of segregating the variable x from the constants. The principle of maintaining equality by performing the same operation on both sides is a fundamental concept in algebra. It ensures that the solution remains valid throughout the manipulation process. This concept is not just a rule; it's a reflection of the balance inherent in mathematical equations.
Further Simplification: Refining the Equation
Simplifying both sides, we get 6x + 18 = 30. Now, we need to isolate the term with x further. We can do this by subtracting 18 from both sides: 6x + 18 - 18 = 30 - 18. This isolates the term with x on the left side. Each subtraction is a calculated step, bringing us closer to the ultimate goal of determining the value of x. This step emphasizes the strategic nature of equation solving. Each operation is chosen deliberately to move towards the desired form of the equation, where x stands alone.
The Final Steps: Unveiling the Solution
This simplifies to 6x = 12. To finally solve for x, we divide both sides by 6: 6x / 6 = 12 / 6. This isolates x completely. The act of division is the final key that unlocks the value of x. It's the culmination of all the previous steps, a direct consequence of the manipulations we've performed. This final step underscores the power of algebraic manipulation in revealing hidden solutions.
The Solution: x = 2
Thus, we find that x = 2. This is the solution to the equation 10x - 3(x - 6) = x + 30. We have successfully navigated the algebraic landscape and arrived at the value of x that satisfies the equation. This final answer is not just a number; it's the culmination of a logical process, a testament to the power of algebraic thinking. The solution represents the precise value that makes the equation true, a value that was initially hidden but revealed through systematic manipulation.
Verification: Ensuring Accuracy
Plugging in the Solution: A Test of Validity
To ensure our solution is correct, we substitute x = 2 back into the original equation: 10(2) - 3(2 - 6) = 2 + 30. This step is crucial for verifying the accuracy of our solution. It's a safeguard against errors that might have crept in during the manipulation process. Verification is not just a formality; it's an integral part of the problem-solving process, a way to build confidence in the result.
Evaluating the Expression: Confirming the Equality
Simplifying, we get 20 - 3(-4) = 32, which further simplifies to 20 + 12 = 32, and finally, 32 = 32. This confirms that our solution is correct. The equality holds, validating our solution and reinforcing the principles of equation solving. This final confirmation is a moment of satisfaction, a reward for the careful and methodical approach we've taken. It underscores the importance of not just finding a solution but also ensuring its validity.
Conclusion: The Power of Algebraic Manipulation
In conclusion, the solution to the equation 10x - 3(x - 6) = x + 30 is indeed x = 2. We arrived at this solution by systematically applying the principles of algebra, including the distributive property, combining like terms, and isolating the variable. The process of solving this equation demonstrates the power and elegance of algebraic manipulation. Each step was a deliberate move, guided by the rules of mathematics, leading us to the correct answer. This journey through the equation not only provides the solution but also reinforces the importance of methodical problem-solving and the beauty of mathematical reasoning. The ability to solve equations is a fundamental skill in mathematics, and this example serves as a stepping stone to tackling more complex problems and appreciating the underlying logic of the mathematical world.
