Solving The Equation 6(x-2)=(1/4)(16x+4) A Step-by-Step Guide

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Unlocking the Secrets of Linear Equations

In the realm of mathematics, solving equations stands as a fundamental skill, a cornerstone upon which more advanced concepts are built. Equation solving isn't just about finding a numerical answer; it's about deciphering the relationships between variables and constants, understanding the language of mathematical expressions, and developing a logical approach to problem-solving. At its core, solving an equation means finding the value(s) of the variable(s) that make the equation true. This often involves a series of algebraic manipulations, guided by the principle of maintaining balance – whatever operation is performed on one side of the equation must also be performed on the other. This ensures that the equality remains valid throughout the solution process. Linear equations, the simplest yet most ubiquitous type of equation, form the bedrock of this mathematical endeavor. They involve variables raised to the power of one, and their solutions are typically straightforward, involving the isolation of the variable on one side of the equation. However, the elegance of linear equations lies not just in their simplicity but also in their widespread applicability. They model a vast array of real-world phenomena, from simple everyday scenarios to complex scientific and engineering problems. Mastering the art of solving linear equations is therefore not just an academic exercise; it's a crucial skill for anyone seeking to understand and interact with the quantitative world around them. The process often involves a combination of techniques, including the distributive property (to eliminate parentheses), combining like terms (to simplify expressions), and applying inverse operations (to isolate the variable). Each step requires careful attention to detail, ensuring that the manipulations are performed correctly and that no errors are introduced along the way. The journey of solving an equation is a journey of logical deduction, a step-by-step unveiling of the hidden value of the variable. With each manipulation, the equation transforms, revealing its underlying structure and guiding us closer to the solution. It is a process that hones our analytical skills, sharpens our understanding of mathematical principles, and empowers us to tackle more complex challenges in the world of numbers.

Deconstructing the Equation: A Step-by-Step Approach

The equation at hand, 6(x - 2) = (1/4)(16x + 4), presents a classic example of a linear equation requiring simplification and solution. To navigate this equation successfully, we'll embark on a methodical journey, breaking down the problem into manageable steps. Our primary objective is to isolate the variable 'x' on one side of the equation, thereby revealing its value. The first step in our quest is to eliminate the parentheses that encase the expressions on both sides of the equation. On the left side, we encounter the expression 6(x - 2). To dispel these parentheses, we invoke the distributive property, a fundamental principle of algebra. This property dictates that we multiply the term outside the parentheses (in this case, 6) by each term inside the parentheses (x and -2). Thus, 6(x - 2) transforms into 6 * x - 6 * 2, which simplifies to 6x - 12. On the right side of the equation, we face a similar challenge with the expression (1/4)(16x + 4). Again, we employ the distributive property, multiplying (1/4) by both 16x and 4. This yields (1/4) * 16x + (1/4) * 4, which simplifies to 4x + 1. Now, our equation has undergone its initial metamorphosis, morphing from 6(x - 2) = (1/4)(16x + 4) into the more streamlined form of 6x - 12 = 4x + 1. We have successfully navigated the first hurdle, paving the way for further simplification. The next phase involves consolidating like terms, a process that brings similar elements together. Our goal is to gather all the 'x' terms on one side of the equation and all the constant terms on the other. This strategic maneuver sets the stage for isolating 'x' and ultimately unveiling its value. To achieve this consolidation, we perform a series of balanced operations, ensuring that the equation remains in equilibrium. We can subtract 4x from both sides of the equation. This action eliminates the 'x' term from the right side, leaving us with only the constant term. However, it's crucial to remember that whatever we do to one side of the equation, we must also do to the other, maintaining the delicate balance of equality. Subtracting 4x from both sides transforms our equation from 6x - 12 = 4x + 1 to 6x - 4x - 12 = 4x - 4x + 1, which simplifies to 2x - 12 = 1. We have successfully corralled the 'x' terms on the left side, bringing us one step closer to our destination.

Isolating the Variable: The Final Push

With the 'x' terms now consolidated on the left side of the equation, our focus shifts to isolating 'x' completely. This involves strategically manipulating the equation to remove any lingering constants that might be hindering our progress. In our current state, the equation stands as 2x - 12 = 1. The constant term -12 is the remaining obstacle separating 'x' from its solitary existence on the left side. To neutralize this -12, we employ the inverse operation – addition. Adding 12 to both sides of the equation will effectively cancel out the -12 on the left, while maintaining the crucial balance of equality. This transformation takes our equation from 2x - 12 = 1 to 2x - 12 + 12 = 1 + 12, which simplifies to 2x = 13. We are now tantalizingly close to our goal. The variable 'x' is almost fully isolated, but it remains tethered to the coefficient 2. To sever this final connection, we again resort to the inverse operation – this time, division. Since 'x' is being multiplied by 2, we divide both sides of the equation by 2. This action will effectively undo the multiplication, leaving 'x' standing alone and revealing its true value. Dividing both sides of 2x = 13 by 2 yields 2x / 2 = 13 / 2, which elegantly simplifies to x = 13/2. At long last, we have arrived at the solution. The variable 'x' has been successfully isolated, and its value revealed to be 13/2. This fraction, also expressible as the decimal 6.5, represents the precise value that, when substituted back into the original equation, will make the equation true. Our journey through the algebraic landscape has culminated in a triumphant discovery, a testament to the power of methodical manipulation and the elegance of mathematical principles.

Solution

Therefore, the solution to the equation 6(x - 2) = (1/4)(16x + 4) is x = 13/2 or x = 6.5.