Solving Linear Equations Step-by-Step Solution For 2.6(5.5p-12.4)=127.92

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Introduction

In the realm of mathematics, linear equations serve as fundamental tools for problem-solving across various disciplines. Mastering the techniques to solve these equations is crucial for anyone venturing into algebra and beyond. This article delves into the step-by-step process of solving a specific linear equation, providing a clear and comprehensive guide suitable for students, educators, and math enthusiasts alike. Our focus is on the equation 2.6(5.5p - 12.4) = 127.92, where our goal is to isolate the variable p and determine its value. We will explore the underlying principles, demonstrate each step meticulously, and offer insights into why these steps are essential. Whether you're brushing up on your algebra skills or tackling linear equations for the first time, this guide will equip you with the knowledge and confidence to solve similar problems effectively. Let's embark on this mathematical journey and unlock the solution to this equation together!

Understanding Linear Equations

Before we dive into the specifics of solving linear equations, let's establish a clear understanding of what they are and why they matter. A linear equation is, at its core, a mathematical statement asserting the equality between two expressions, where the variable (in our case, p) appears with a power of one. This means there are no exponents, square roots, or other complex operations applied to the variable. The equation, when graphed, produces a straight line, hence the term "linear." These equations are ubiquitous in mathematics and find applications in physics, engineering, economics, and computer science, among other fields.

The significance of linear equations lies in their ability to model real-world phenomena in a simplified yet accurate manner. They allow us to represent relationships between quantities and make predictions based on these relationships. For example, linear equations can describe the motion of an object at constant velocity, the relationship between supply and demand in economics, or the flow of electricity in a circuit. Understanding how to solve linear equations is, therefore, not just an academic exercise but a practical skill that empowers us to analyze and interpret the world around us.

The key to solving a linear equation is to isolate the variable on one side of the equation. This involves performing a series of algebraic operations that maintain the equality while simplifying the equation until the variable is alone. These operations include addition, subtraction, multiplication, and division, and they must be applied consistently to both sides of the equation to preserve the balance. In the following sections, we will apply these principles to solve the equation 2.6(5.5p - 12.4) = 127.92, demonstrating each step with clarity and precision. Understanding the fundamental nature of linear equations sets the stage for a deeper appreciation of the solution process.

Step-by-Step Solution

To effectively solve the linear equation 2.6(5.5p - 12.4) = 127.92, we will follow a systematic approach, breaking down each step to ensure clarity and comprehension. This process involves distributing, simplifying, isolating the variable, and verifying the solution. Let's embark on this step-by-step journey.

1. Distribute

The initial step in solving the equation is to eliminate the parentheses by distributing the 2.6 across the terms inside the parenthesis. This involves multiplying 2.6 by both 5.5p and -12.4. This application of the distributive property transforms the equation into a more manageable form. Mathematically, the distribution looks like this:

  1. 6 * 5.5p - 2.6 * 12.4 = 127.92
  2. 3p - 32.24 = 127.92

This step is crucial because it removes the grouping, allowing us to combine like terms and eventually isolate p. The result, 14.3p - 32.24 = 127.92, sets the stage for the subsequent steps in solving the equation.

2. Isolate the Term with p

Our next objective is to isolate the term containing p, which is 14.3p. To achieve this, we need to eliminate the constant term, -32.24, from the left side of the equation. We accomplish this by adding 32.24 to both sides of the equation. Adding the same value to both sides maintains the balance of the equation, a fundamental principle in algebra. The process is as follows:

  1. 3p - 32.24 + 32.24 = 127.92 + 32.24
  2. 3p = 160.16

By adding 32.24 to both sides, we have successfully isolated the term with p on the left side, resulting in the simplified equation 14.3p = 160.16. This step brings us closer to our goal of determining the value of p.

3. Solve for p

Now that we have isolated the term 14.3p, the final step in solving for p is to divide both sides of the equation by the coefficient of p, which is 14.3. Division, like addition, subtraction, and multiplication, must be applied to both sides of the equation to maintain equality. This process will leave p isolated on one side, revealing its value. The calculation is as follows:

p = 160.16 / 14.3

p = 11.2

Therefore, by dividing both sides by 14.3, we find that p equals 11.2. This is the solution to the equation. We have successfully navigated the steps of solving a linear equation, arriving at the value of p.

4. Verification

To ensure the accuracy of our solution, it is essential to verify that p = 11.2 indeed satisfies the original equation. We achieve this by substituting 11.2 for p in the equation 2.6(5.5p - 12.4) = 127.92 and checking if the left side equals the right side.

Let's substitute p = 11.2 into the equation:

  1. 6(5.5 * 11.2 - 12.4) = 127.92
  2. 6(61.6 - 12.4) = 127.92
  3. 6(49.2) = 127.92
  4. 92 = 127.92

Since the left side equals the right side, our solution p = 11.2 is correct. This verification step provides confidence in our result and reinforces the understanding of the solution process.

Conclusion

In conclusion, we have successfully solved the linear equation 2.6(5.5p - 12.4) = 127.92 by following a clear, step-by-step approach. We began by distributing the 2.6 across the terms inside the parentheses, which simplified the equation. Next, we isolated the term containing p by adding 32.24 to both sides of the equation. We then solved for p by dividing both sides by 14.3, arriving at the solution p = 11.2. Finally, we verified our solution by substituting 11.2 back into the original equation, confirming its accuracy. This process underscores the importance of each step in solving linear equations and highlights the principles of maintaining equality throughout the algebraic manipulations.

Solving linear equations is a fundamental skill in mathematics, with applications extending far beyond the classroom. It is a cornerstone of problem-solving in various fields, including science, engineering, economics, and computer science. Mastering these techniques not only enhances mathematical proficiency but also cultivates critical thinking and analytical skills that are valuable in numerous contexts.

By understanding the underlying principles and practicing consistently, anyone can become adept at solving linear equations. This guide provides a solid foundation for tackling more complex mathematical problems and serves as a testament to the power and elegance of algebra. As you continue your mathematical journey, remember that each equation solved is a step forward in your understanding and a testament to your problem-solving abilities. Keep practicing, keep exploring, and embrace the challenges that mathematics presents, for they are the keys to unlocking a deeper understanding of the world around us.