Solving The Equation (1/3)x + 4 = 10 A Step-by-Step Guide
Introduction: Unveiling the Mystery of Linear Equations
In the realm of mathematics, linear equations stand as fundamental building blocks, providing a framework for understanding relationships between variables. At their core, linear equations express a connection where the highest power of the variable is one, resulting in a straight-line graph. Solving linear equations is a crucial skill, enabling us to determine the unknown value (often represented as 'x') that satisfies the equation's condition. This article will embark on a journey to dissect and conquer the linear equation (1/3)x + 4 = 10, providing a clear roadmap for finding the elusive value of x.
Linear equations, while seemingly simple, have profound applications across various fields. From physics and engineering to economics and computer science, linear models serve as powerful tools for representing and solving real-world problems. Whether calculating the trajectory of a projectile, predicting market trends, or optimizing resource allocation, the principles of linear equations form the bedrock of analytical thinking. Understanding how to manipulate and solve these equations equips us with the ability to decipher complex scenarios and arrive at informed decisions. The process of solving a linear equation involves isolating the variable of interest, effectively unraveling the equation's structure to reveal the solution. This often requires a series of algebraic manipulations, guided by the principles of equality and inverse operations. By systematically applying these techniques, we can navigate through the equation, step by step, until the variable stands alone, exposing its true value. In the following sections, we will delve into the specific steps required to solve the equation (1/3)x + 4 = 10, transforming a seemingly daunting task into a manageable and rewarding exercise.
Dissecting the Equation: (1/3)x + 4 = 10
The equation at hand, (1/3)x + 4 = 10, presents a classic linear equation structure. It comprises a variable term ((1/3)x), a constant term (+4), and an equality sign (=) that bridges the left-hand side (LHS) and the right-hand side (RHS). Our mission is to isolate 'x' on one side of the equation, effectively revealing its value. To achieve this, we will employ a series of strategic steps, each designed to chip away at the equation's complexity. The first step in solving this equation involves addressing the constant term (+4) on the LHS. Our goal is to eliminate this term, paving the way for isolating the variable term ((1/3)x). To accomplish this, we invoke the principle of inverse operations. Since the constant term is added to the variable term, we perform the inverse operation – subtraction. Subtracting 4 from both sides of the equation maintains the balance, ensuring that the equality remains intact. This crucial step allows us to simplify the equation, bringing us closer to our objective of isolating 'x'. The next step in our quest to isolate 'x' involves tackling the coefficient (1/3) that multiplies the variable. This coefficient represents a fractional scaling of 'x', and our aim is to undo this scaling. To achieve this, we again rely on the principle of inverse operations. Since 'x' is multiplied by (1/3), we perform the inverse operation – multiplication by the reciprocal. The reciprocal of (1/3) is 3, so we multiply both sides of the equation by 3. This strategic maneuver effectively cancels out the fractional coefficient, leaving 'x' standing alone on the LHS. With 'x' isolated, the equation now reveals its solution, providing the numerical value that satisfies the original equation.
Step-by-Step Solution: A Journey to Isolate x
Let's embark on a step-by-step journey to solve the equation (1/3)x + 4 = 10. This meticulous approach will demonstrate how to isolate 'x' and uncover its value.
Step 1: Isolating the Variable Term
Our initial objective is to isolate the variable term ((1/3)x) on the left-hand side (LHS) of the equation. To achieve this, we need to eliminate the constant term (+4) that is currently hindering our progress. The principle of inverse operations dictates that we perform the opposite operation to undo the addition. In this case, we subtract 4 from both sides of the equation. This ensures that the equation remains balanced, preserving the equality between the LHS and the right-hand side (RHS).
(1/3)x + 4 - 4 = 10 - 4
Simplifying both sides, we obtain:
(1/3)x = 6
Step 2: Eliminating the Coefficient
Now that we have isolated the variable term, our next challenge is to eliminate the coefficient (1/3) that multiplies 'x'. This coefficient represents a fractional scaling of 'x', and we need to undo this scaling to reveal the true value of 'x'. Again, we turn to the principle of inverse operations. Since 'x' is multiplied by (1/3), we perform the inverse operation – multiplication by the reciprocal. The reciprocal of (1/3) is 3, so we multiply both sides of the equation by 3. This strategic maneuver effectively cancels out the fractional coefficient, leaving 'x' standing alone on the LHS.
3 * (1/3)x = 3 * 6
Simplifying both sides, we arrive at:
x = 18
The Solution: x = 18
Through this meticulous step-by-step process, we have successfully isolated 'x' and determined its value. The solution to the equation (1/3)x + 4 = 10 is x = 18. This value satisfies the original equation, making the LHS equal to the RHS. To verify our solution, we can substitute x = 18 back into the original equation and check if the equality holds true. This process of verification provides confidence in our solution and reinforces our understanding of the equation's behavior. In the following section, we will delve into the verification process, solidifying our understanding of the solution and the principles that underpin it.
Verifying the Solution: Ensuring Accuracy and Understanding
To ensure the accuracy of our solution and deepen our understanding of the equation, we must verify the solution x = 18. This process involves substituting the obtained value of 'x' back into the original equation and checking if the equality holds true. If the left-hand side (LHS) of the equation equals the right-hand side (RHS) after the substitution, our solution is validated. This verification step serves as a crucial safeguard against errors, providing confidence in our answer. Furthermore, it reinforces our grasp of the equation's behavior and the relationship between its variables and constants.
The Verification Process
Let's substitute x = 18 into the original equation:
(1/3)x + 4 = 10
Replacing 'x' with 18, we get:
(1/3)(18) + 4 = 10
Simplifying the expression, we first evaluate the multiplication:
6 + 4 = 10
Next, we perform the addition:
10 = 10
The Result: Equality Confirmed
The result of our substitution and simplification is a resounding confirmation of our solution's accuracy. The LHS of the equation, after substituting x = 18, equals the RHS, demonstrating that the equality holds true. This verification step not only validates our solution but also reinforces our understanding of the equation's underlying structure. By substituting the solution back into the original equation, we effectively retrace our steps, ensuring that each manipulation and simplification was performed correctly. This process fosters a deeper appreciation for the equation's behavior and the principles of algebraic manipulation. In the concluding section, we will summarize our journey, highlighting the key steps involved in solving linear equations and emphasizing the importance of verification in mathematical problem-solving.
Conclusion: Mastering Linear Equations and the Power of Verification
In this comprehensive exploration, we have successfully navigated the realm of linear equations, mastering the art of solving for the unknown variable 'x'. Our journey began with a dissection of the equation (1/3)x + 4 = 10, identifying its components and outlining our objective – to isolate 'x'. We then embarked on a step-by-step solution, employing the principles of inverse operations to strategically manipulate the equation. By subtracting the constant term and multiplying by the reciprocal of the coefficient, we meticulously peeled away the layers of complexity, ultimately revealing the solution: x = 18. However, our journey did not conclude with the determination of the solution. Recognizing the paramount importance of accuracy and understanding, we engaged in a crucial verification process. Substituting x = 18 back into the original equation, we meticulously checked if the equality held true. The resounding confirmation, 10 = 10, validated our solution, instilling confidence in our answer and reinforcing our comprehension of the equation's behavior. This emphasis on verification underscores a fundamental principle in mathematics – the pursuit of accuracy and the critical evaluation of results.
Solving linear equations is not merely a mechanical exercise; it is a journey of logical deduction and strategic manipulation. By mastering the principles of inverse operations and the importance of maintaining equality, we can confidently tackle a wide range of linear equations. Moreover, the habit of verifying solutions instills a culture of precision and critical thinking, essential qualities in mathematical problem-solving. As we conclude this exploration, we carry with us not only the solution to a specific equation but also a deeper appreciation for the power and elegance of linear equations and the transformative impact of rigorous verification.