Solving The Equation 71 + 1 - [(3X + 4) - (4X + 3)] = 25 - [-5X + (-2x + 3)] A Step-by-Step Guide

Introduction

In the realm of mathematics, equations serve as fundamental tools for expressing relationships and solving for unknown variables. This article delves into the intricate details of solving the equation 71 + 1 - [(3X + 4) - (4X + 3)] = 25 - [-5X + (-2x + 3)]. We will meticulously dissect the equation, employing algebraic principles and techniques to arrive at the solution for the variable 'X'. This journey will not only provide a step-by-step solution but also enhance your understanding of algebraic manipulations and problem-solving strategies. Whether you're a student grappling with algebra or simply a math enthusiast, this comprehensive guide will illuminate the path to unraveling this equation.

Dissecting the Equation: A Step-by-Step Approach

1. Simplifying Expressions Within Parentheses

The initial step in solving any complex equation involves simplifying the expressions enclosed within parentheses. This not only declutters the equation but also paves the way for subsequent operations. Let's begin by tackling the inner parentheses within our equation: 71 + 1 - [(3X + 4) - (4X + 3)] = 25 - [-5X + (-2x + 3)].

Consider the expression (3X + 4) - (4X + 3). To simplify this, we distribute the negative sign to the second term, resulting in 3X + 4 - 4X - 3. Combining like terms, we get -X + 1. This simplification reduces the complexity of the left-hand side of the equation. Similarly, on the right-hand side, we address the expression -5X + (-2x + 3). Here, we can directly combine the terms involving 'X', leading to -7X + 3. By simplifying these expressions within parentheses, we've taken the first significant step towards solving the equation.

2. Further Simplification and Distribution

With the inner parentheses simplified, we now focus on the outer structures. The equation now looks like this: 71 + 1 - [-X + 1] = 25 - [-7X + 3]. The next logical step is to distribute the negative signs preceding the brackets on both sides of the equation. On the left-hand side, we have - [-X + 1]. Distributing the negative sign, we get X - 1. Applying this to the left side of the equation gives us 71 + 1 + X - 1. Similarly, on the right-hand side, we have - [-7X + 3]. Distributing the negative sign yields 7X - 3. Incorporating this into the right side of the equation, we get 25 + 7X - 3. This process of distributing negative signs is crucial for accurately simplifying and solving equations involving multiple parentheses and brackets.

3. Combining Like Terms

Now that we've eliminated the parentheses and brackets, the equation becomes significantly more manageable. Our equation now stands as 71 + 1 + X - 1 = 25 + 7X - 3. The next logical step is to combine like terms on both sides of the equation. On the left-hand side, we have the constant terms 71, 1, and -1. Combining these, we get 71. Thus, the left side simplifies to 71 + X. On the right-hand side, we have the constant terms 25 and -3. Combining these, we get 22. So, the right side simplifies to 22 + 7X. By combining like terms, we've condensed the equation into a more concise and easier-to-handle form. This step is essential for isolating the variable 'X' and eventually finding its value.

4. Isolating the Variable 'X'

After simplifying both sides of the equation, we arrive at 71 + X = 22 + 7X. The primary goal now is to isolate the variable 'X' on one side of the equation. To achieve this, we need to manipulate the equation in a way that groups all terms containing 'X' on one side and all constant terms on the other. A common strategy is to subtract the term with the smaller coefficient of 'X' from both sides. In this case, we subtract 'X' from both sides, resulting in 71 = 22 + 6X. This move eliminates 'X' from the left side, bringing us closer to isolating it. Next, we need to isolate the term with 'X' by removing the constant term on the same side. We subtract 22 from both sides, which gives us 49 = 6X. This step effectively isolates the term containing 'X', making it easier to solve for the variable itself.

5. Solving for 'X'

Having isolated the term 6X, the final step is to solve for 'X'. The equation now reads 49 = 6X. To find the value of 'X', we need to undo the multiplication by 6. This is achieved by dividing both sides of the equation by 6. Performing this operation, we get X = 49/6. This fraction represents the value of 'X' that satisfies the original equation. While it can be left as an improper fraction, it can also be expressed as a mixed number or a decimal, depending on the context or the desired level of precision. Thus, by dividing both sides by the coefficient of 'X', we successfully solved for the variable and found its value.

Validating the Solution

1. Substituting the Value of 'X'

After solving for 'X', it's crucial to validate the solution by substituting the obtained value back into the original equation. This step ensures that our solution is accurate and that no errors were made during the simplification process. Our original equation is 71 + 1 - [(3X + 4) - (4X + 3)] = 25 - [-5X + (-2x + 3)], and we found that X = 49/6. Substituting X = 49/6 into the equation involves replacing every instance of 'X' with 49/6.

This substitution transforms the equation into a numerical expression that we can evaluate. We first substitute 49/6 for 'X' on both sides of the equation. Then, we meticulously perform the arithmetic operations, following the order of operations (PEMDAS/BODMAS). This involves simplifying expressions within parentheses, performing multiplications and divisions, and finally, additions and subtractions. If the left-hand side of the equation equals the right-hand side after this substitution and simplification, our solution is correct.

2. Verifying Equality

Once we've substituted the value of X = 49/6 into the original equation, the next step is to verify that both sides of the equation are indeed equal. This involves performing all the necessary arithmetic operations on both sides and comparing the results. Let's break down the process step by step.

First, we substitute X = 49/6 into the equation: 71 + 1 - [(3(49/6) + 4) - (4(49/6) + 3)] = 25 - [-5*(49/6) + (-2*(49/6) + 3)]**. Next, we simplify the expressions within the parentheses. This involves multiplying fractions, adding and subtracting, and combining like terms. It's crucial to follow the order of operations carefully to avoid errors. After simplifying the expressions within the parentheses, we move on to the brackets, again performing the necessary arithmetic operations. Finally, we perform the remaining additions and subtractions on both sides of the equation. If, after all these calculations, the left-hand side of the equation is equal to the right-hand side, we can confidently conclude that our solution, X = 49/6, is correct. This verification process is a critical safeguard against errors and ensures the accuracy of our solution.

Alternative Approaches to Solving the Equation

1. Graphical Method

While algebraic methods are the most common way to solve equations, graphical methods offer a visual approach that can be particularly insightful. In the case of the equation 71 + 1 - [(3X + 4) - (4X + 3)] = 25 - [-5X + (-2x + 3)], we can employ a graphical method by treating each side of the equation as a separate function. Let's denote the left-hand side as f(X) = 71 + 1 - [(3X + 4) - (4X + 3)] and the right-hand side as g(X) = 25 - [-5X + (-2x + 3)].

To proceed graphically, we plot these two functions, f(X) and g(X), on the same coordinate plane. Each function will be represented by a line, and the point(s) where these lines intersect represent the solution(s) to the equation. The x-coordinate of the intersection point(s) corresponds to the value(s) of 'X' that satisfy the equation. This graphical method provides a visual confirmation of the solution obtained algebraically. It's particularly useful for understanding the nature of the solutions and for equations that may have multiple solutions or no solutions. By visualizing the equation as two intersecting lines, we gain a different perspective on the problem and its solution.

2. Using Computer Algebra Systems (CAS)

In today's technological age, Computer Algebra Systems (CAS) offer powerful tools for solving complex equations with ease and precision. CAS software, such as Mathematica, Maple, or even online tools like Wolfram Alpha, can handle intricate algebraic manipulations and provide solutions in various forms. To solve the equation 71 + 1 - [(3X + 4) - (4X + 3)] = 25 - [-5X + (-2x + 3)] using a CAS, one simply needs to input the equation into the system.

The CAS will then process the equation, applying algebraic rules and techniques to isolate the variable 'X' and determine its value. The software can not only provide the solution but also display the intermediate steps, offering a detailed breakdown of the solving process. This is particularly beneficial for learning and understanding the underlying algebraic principles. CAS tools are invaluable for handling complex equations that may be time-consuming or prone to errors when solved manually. They also allow for quick verification of solutions obtained through manual methods. By leveraging CAS, we can tackle challenging equations with confidence and efficiency.

Common Mistakes to Avoid

1. Incorrect Distribution of Signs

A common pitfall in solving algebraic equations, especially those involving parentheses and negative signs, is the incorrect distribution of signs. This often occurs when dealing with expressions like - (a - b), where the negative sign must be distributed to both 'a' and '-b'. The correct distribution yields -a + b, but a frequent mistake is to apply the negative sign only to the first term, resulting in -a - b. In the context of our equation, 71 + 1 - [(3X + 4) - (4X + 3)] = 25 - [-5X + (-2x + 3)], incorrect sign distribution can occur when simplifying the expressions within the brackets. For instance, when removing the parentheses in - [(3X + 4) - (4X + 3)], it's crucial to distribute the negative sign to both terms inside the square brackets. Failing to do so can lead to an incorrect equation and, consequently, an incorrect solution. To avoid this, always double-check the distribution of signs, especially when dealing with multiple levels of parentheses and brackets. It's a small step that can significantly impact the accuracy of the solution.

2. Combining Non-Like Terms

Another prevalent error in algebraic manipulations is the combination of non-like terms. Like terms are those that have the same variable raised to the same power. For example, 3X and -5X are like terms because they both involve the variable 'X' raised to the power of 1. Similarly, 7 and -2 are like terms as they are both constants. However, 3X and 7 are not like terms and cannot be combined directly. In the process of simplifying equations, it's essential to combine only like terms. For example, in our equation, 71 + 1 - [(3X + 4) - (4X + 3)] = 25 - [-5X + (-2x + 3)], we can combine the constant terms 71 and 1 on the left-hand side and the 'X' terms -5X and -2x on the right-hand side. However, we cannot combine a constant term with an 'X' term. Attempting to do so will result in an incorrect simplification and lead to an incorrect solution. To avoid this mistake, always carefully identify like terms before combining them, ensuring that they have the same variable and exponent.

Real-World Applications of Algebraic Equations

1. Engineering and Physics

Algebraic equations serve as the bedrock of numerous calculations and problem-solving techniques. In engineering, algebraic equations are used to model and analyze structures, circuits, and systems. For instance, engineers use equations to determine the forces acting on a bridge, the flow of electricity in a circuit, or the dynamics of a mechanical system. These equations often involve multiple variables and require careful manipulation to solve for the desired quantities. Similarly, in physics, algebraic equations are essential for describing motion, energy, and forces. From calculating the trajectory of a projectile to determining the energy of a particle, algebraic equations provide the framework for understanding and predicting physical phenomena. The ability to solve algebraic equations is thus a fundamental skill for engineers and physicists, enabling them to design, analyze, and innovate in their respective fields. The accuracy and efficiency with which these equations are solved directly impact the success and safety of engineering projects and the validity of physics-based predictions.

2. Economics and Finance

Algebraic equations are indispensable tools for modeling and analyzing economic and financial phenomena. Economists use equations to represent supply and demand curves, macroeconomic models, and various economic relationships. These equations help in forecasting economic trends, understanding market behavior, and evaluating policy impacts. For instance, algebraic equations are used to determine equilibrium prices, calculate inflation rates, and model economic growth. In finance, algebraic equations are crucial for calculating investment returns, assessing risk, and valuing assets. Financial analysts use equations to determine the present value of future cash flows, calculate loan payments, and model portfolio performance. The ability to manipulate and solve algebraic equations is thus essential for professionals in economics and finance. Accurate solutions to these equations can inform critical decisions, from investment strategies to government policies, highlighting the real-world significance of algebraic skills.

Conclusion

In this comprehensive exploration, we have meticulously solved the equation 71 + 1 - [(3X + 4) - (4X + 3)] = 25 - [-5X + (-2x + 3)], demonstrating the step-by-step process of simplifying, isolating the variable, and validating the solution. We've also delved into alternative solution methods, such as graphical approaches and the use of Computer Algebra Systems, offering a broader perspective on problem-solving strategies. Furthermore, we've highlighted common pitfalls to avoid, emphasizing the importance of careful sign distribution and the combination of like terms. Finally, we've underscored the real-world applications of algebraic equations in diverse fields like engineering, physics, economics, and finance, showcasing the practical relevance of mastering these mathematical skills. By understanding the intricacies of algebraic manipulations and problem-solving techniques, individuals can enhance their analytical abilities and tackle a wide range of challenges in both academic and professional settings.