Solving The Equation 2(x+4)=70 A Step-by-Step Guide

Understanding the Basics of Algebraic Equations

In the realm of mathematics, algebraic equations form the bedrock of problem-solving and analytical thinking. These equations, often represented as expressions with variables, constants, and mathematical operations, provide a framework for understanding relationships between quantities and finding unknown values. In this comprehensive guide, we will delve into the intricacies of solving the equation 2(x+4)=70, providing a step-by-step explanation that will not only illuminate the solution but also solidify your understanding of fundamental algebraic principles. Grasping these concepts is crucial as they serve as building blocks for more advanced mathematical topics and real-world applications.

At its core, an algebraic equation is a statement of equality between two expressions. The goal is to isolate the variable, in this case, 'x', to determine its value. This involves applying various mathematical operations while maintaining the balance of the equation. For instance, the equation 2(x+4)=70 presents a scenario where 'x' is an unknown quantity that we aim to find. To achieve this, we will employ techniques such as the distributive property, inverse operations, and simplification to gradually peel away the layers surrounding 'x' until we arrive at its true value. The journey of solving this equation is not merely about finding the answer; it is about honing your logical reasoning and problem-solving skills, which are indispensable tools in mathematics and beyond. As we embark on this step-by-step exploration, remember that each operation performed is a deliberate move designed to bring us closer to the solution, and understanding the rationale behind each step is paramount to mastering algebraic equations.

The Distributive Property: Expanding the Equation

Our journey to solve the equation 2(x+4)=70 begins with a crucial step: applying the distributive property. This fundamental principle of algebra allows us to simplify expressions that involve parentheses. In essence, the distributive property states that multiplying a number by a sum (or difference) is the same as multiplying the number by each term inside the parentheses individually and then adding (or subtracting) the results. This property is like a key that unlocks the equation, allowing us to separate the terms and make them more manageable.

In our case, we have the term 2(x+4). To apply the distributive property, we multiply the '2' by both 'x' and '4'. This yields 2 * x + 2 * 4, which simplifies to 2x + 8. By expanding the expression within the parentheses, we transform the equation from 2(x+4)=70 to 2x + 8 = 70. This seemingly small step is significant because it disentangles 'x' from the parentheses, paving the way for further simplification. The distributive property is not just a mathematical rule; it is a strategic tool that empowers us to manipulate equations and make them more amenable to solving. As we progress through the solution, we will see how this initial step sets the stage for the subsequent operations that will ultimately reveal the value of 'x'. Understanding and mastering the distributive property is, therefore, a cornerstone of algebraic proficiency.

Isolating the Variable: The Subtraction Step

With the equation now in the form 2x + 8 = 70, our next objective is to isolate the term containing the variable 'x'. This is a critical step in solving for 'x', as it brings us closer to determining its value. To achieve this isolation, we employ the concept of inverse operations. In this case, we have the term '+ 8' on the same side as '2x'. The inverse operation of addition is subtraction, so we subtract 8 from both sides of the equation. This is a fundamental principle of algebraic manipulation: to maintain the balance of the equation, any operation performed on one side must also be performed on the other.

Subtracting 8 from both sides of 2x + 8 = 70 gives us 2x + 8 - 8 = 70 - 8. Simplifying this, we get 2x = 62. Notice how the '+ 8' on the left side has been effectively cancelled out, leaving the term '2x' isolated. This step demonstrates the power of inverse operations in unraveling equations. By strategically applying the appropriate inverse operation, we can systematically eliminate terms that hinder our progress towards isolating the variable. This subtraction step is not just a mechanical process; it is a deliberate maneuver guided by the principle of maintaining equality and the goal of isolating 'x'. As we continue to solve the equation, we will see how each step builds upon the previous one, gradually revealing the solution.

The Final Step: Division to Solve for x

Having successfully isolated the term '2x' in the equation 2x = 62, we are now on the verge of solving for 'x'. The final step involves undoing the multiplication that is currently binding 'x'. In this case, 'x' is being multiplied by 2. To isolate 'x', we need to perform the inverse operation of multiplication, which is division. Just as with the subtraction step, we must maintain the balance of the equation by performing the same operation on both sides.

Therefore, we divide both sides of the equation 2x = 62 by 2. This gives us 2x / 2 = 62 / 2. Simplifying this, we get x = 31. This is the culmination of our step-by-step journey, the final answer that reveals the value of the unknown variable. The division step is the last piece of the puzzle, the act that liberates 'x' and allows us to see its true value. This process underscores the elegance and precision of algebra, where each operation serves a specific purpose in the quest to solve for unknowns. With 'x' now isolated and its value determined, we have successfully navigated the equation 2(x+4)=70, demonstrating the power of algebraic principles and the methodical approach to problem-solving.

Verification: Ensuring the Solution's Accuracy

Before declaring victory in solving the equation 2(x+4)=70, it is crucial to verify our solution. This step is not merely a formality; it is a vital safeguard against errors and a testament to the rigor of mathematical problem-solving. Verification involves substituting the value we found for 'x' back into the original equation and checking if the equation holds true. This process acts as a litmus test, confirming whether our step-by-step manipulations have led us to the correct answer.

In our case, we found that x = 31. To verify this, we substitute 31 for 'x' in the original equation, 2(x+4)=70. This gives us 2(31+4)=70. Simplifying the expression inside the parentheses, we get 2(35)=70. Performing the multiplication, we arrive at 70=70. This result confirms that our solution, x = 31, is indeed correct. The left side of the equation equals the right side, validating our step-by-step process. Verification not only provides assurance in the correctness of the answer but also reinforces the understanding of the equation's structure and the relationships between its components. It is an indispensable practice that transforms the act of solving equations from a mere mechanical exercise into a confident and validated conclusion.

Conclusion: Mastering Algebraic Equations

In this comprehensive exploration, we have meticulously dissected the equation 2(x+4)=70, demonstrating a step-by-step approach to solving for the unknown variable 'x'. From the initial application of the distributive property to the final act of verification, each step was carefully explained, highlighting the underlying algebraic principles at play. We saw how the distributive property allows us to expand expressions and disentangle variables from parentheses. We then employed inverse operations, strategically subtracting and dividing to isolate 'x' and ultimately determine its value. The final verification step underscored the importance of accuracy and confirmed the validity of our solution.

This journey through a single equation encapsulates the essence of algebraic problem-solving. It is a process that demands not only technical skill but also logical reasoning and a deep understanding of mathematical concepts. Mastering algebraic equations is not just about finding answers; it is about developing a mindset of analytical thinking and problem-solving that extends far beyond the realm of mathematics. The ability to break down complex problems into manageable steps, apply appropriate techniques, and verify the results is a skill that is invaluable in various aspects of life. As you continue your mathematical journey, remember that each equation solved is a step forward in honing your problem-solving prowess and expanding your intellectual horizons. Embrace the challenges, celebrate the solutions, and continue to explore the fascinating world of mathematics.