Simplifying 3a + 2(ab + C) = 2a A Step-by-Step Guide

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In the realm of mathematics, equations serve as the cornerstone of problem-solving, representing relationships between variables and constants. Among these equations, linear equations hold a prominent position, renowned for their simplicity and wide applicability. Today, we embark on a journey to unravel the intricacies of simplifying a specific linear equation: 3a + 2(ab + c) = 2a. This equation, seemingly straightforward at first glance, presents a unique opportunity to delve into the fundamental principles of algebraic manipulation, ultimately leading us to a more concise and manageable form.

Decoding the Equation: A Step-by-Step Approach

Our primary objective is to isolate the variable 'a' on one side of the equation, allowing us to gain a clearer understanding of its relationship with the other variables and constants. To achieve this, we will employ a series of algebraic techniques, each meticulously chosen to preserve the equation's integrity while gradually simplifying its structure.

1. Distributive Property: Expanding the Equation

The first step in our simplification journey involves applying the distributive property, a fundamental principle that allows us to expand expressions enclosed within parentheses. In our equation, the term 2(ab + c) beckons for expansion. By distributing the '2' across the terms within the parentheses, we transform the expression into 2ab + 2c. This seemingly simple maneuver unlocks the equation's hidden potential, paving the way for further simplification.

Our equation now takes on a new form: 3a + 2ab + 2c = 2a. The once-compact term has blossomed into a more intricate expression, revealing the individual components that contribute to the equation's overall structure. This expansion is crucial, as it allows us to identify like terms, which we will later combine to further simplify the equation.

2. Combining Like Terms: Streamlining the Equation

With the distributive property successfully applied, we now turn our attention to identifying and combining like terms. Like terms, in the realm of algebra, are terms that share the same variable raised to the same power. In our equation, we observe two terms containing the variable 'a': 3a and 2a. These terms, being of the same breed, can be combined to streamline the equation.

To combine these terms, we simply add their coefficients, the numerical values that precede the variables. In this case, we add the coefficients 3 and 2, resulting in 5. Thus, the terms 3a and 2a coalesce into a single term: 5a. Our equation now stands as: 5a + 2ab + 2c = 2a. The equation is gradually taking shape, shedding its initial complexity as we systematically apply algebraic principles.

3. Isolating the Variable: A Quest for Clarity

Our ultimate goal is to isolate the variable 'a' on one side of the equation, thereby revealing its relationship with the other variables and constants. To achieve this, we must strategically move terms across the equality sign, always maintaining the equation's balance. In our current equation, we observe the term 2a on the right-hand side, which we aim to eliminate from that side.

To eliminate 2a from the right-hand side, we employ the principle of inverse operations. We subtract 2a from both sides of the equation, ensuring that the equation remains balanced. This maneuver effectively cancels out the 2a on the right-hand side, leaving us with: 5a + 2ab + 2c - 2a = 0. The equation is gradually converging towards its simplified form, as we strategically isolate the variable of interest.

4. Further Simplification: Unveiling the Equation's Essence

With the 2a term successfully banished from the right-hand side, we can further simplify the equation by combining the remaining 'a' terms on the left-hand side. We have 5a and -2a, which, when combined, yield 3a. Our equation now proudly stands as: 3a + 2ab + 2c = 0. The equation is nearing its final simplified form, its essence gradually unveiled through our meticulous algebraic manipulations.

5. The Final Flourish: Expressing 'a' in Terms of Others

Our quest to simplify the equation has led us to a point where we can express the variable 'a' in terms of the other variables, 'b' and 'c'. To achieve this, we isolate the terms containing 'a' on one side of the equation, while moving the remaining terms to the other side.

We begin by subtracting 2c from both sides of the equation, effectively moving the constant term to the right-hand side: 3a + 2ab = -2c. Next, we factor out 'a' from the left-hand side, revealing the common factor that binds the terms: a(3 + 2b) = -2c. Finally, to isolate 'a', we divide both sides of the equation by (3 + 2b), resulting in: a = -2c / (3 + 2b). The variable 'a' now stands alone, expressed in terms of 'b' and 'c', providing a clear understanding of their relationship.

The Simplified Equation: A Triumph of Algebraic Manipulation

Through a series of meticulous algebraic steps, we have successfully simplified the equation 3a + 2(ab + c) = 2a, arriving at the elegant expression: a = -2c / (3 + 2b). This simplified form provides a clearer understanding of the relationship between the variables 'a', 'b', and 'c'.

Importance of Simplification: Unveiling Clarity and Efficiency

The simplification of equations is not merely an academic exercise; it holds profound practical implications. Simplified equations are easier to comprehend, manipulate, and solve. They allow us to gain insights into the underlying relationships between variables and constants, paving the way for informed decision-making in various fields.

In mathematics, simplification is a fundamental skill, essential for tackling complex problems. It allows us to break down intricate equations into manageable components, revealing their hidden structure and paving the way for solutions. Simplified equations are also crucial in scientific and engineering applications, where they are used to model real-world phenomena and make predictions.

Conclusion: A Journey of Mathematical Discovery

Our journey through the simplification of the equation 3a + 2(ab + c) = 2a has been a testament to the power of algebraic manipulation. We have witnessed how a seemingly complex equation can be transformed into a more concise and understandable form through the application of fundamental principles.

Simplification is not just about finding the right answer; it is about developing a deeper understanding of the underlying mathematical relationships. It is about unraveling the complexities and revealing the elegance that lies within the world of equations. As we continue our mathematical explorations, let us embrace the art of simplification, for it is the key to unlocking a world of knowledge and discovery.

  • Simplifying Equations
  • Algebraic Manipulation
  • Linear Equations
  • Distributive Property
  • Combining Like Terms
  • Isolating Variables
  • Solving Equations
  • Mathematical Simplification
  • Equation Transformation
  • Algebraic Techniques