Solving For P In The Linear Equation 24p + 12 - 18p = 10 + 2p - 6
Introduction
In the realm of mathematics, linear equations serve as fundamental tools for modeling and solving a wide array of problems. These equations, characterized by variables raised to the first power, offer a straightforward yet powerful way to represent relationships between quantities. This article delves into the process of solving a specific linear equation, 24p + 12 - 18p = 10 + 2p - 6, to determine the value of the variable 'p'. Understanding how to manipulate and solve such equations is crucial for anyone venturing into the fields of algebra, calculus, and beyond. Linear equations form the bedrock of numerous mathematical concepts, and mastering their solution is a pivotal step in developing mathematical proficiency. From simple word problems to complex scientific models, the ability to solve for unknowns in linear relationships is an indispensable skill. In this exploration, we will dissect the equation step by step, employing algebraic principles to isolate 'p' and arrive at its definitive value. The journey through this equation will not only reveal the solution but also reinforce the core techniques involved in solving linear equations, enhancing your problem-solving capabilities in mathematics and related disciplines.
Understanding Linear Equations
Before diving into the solution, let's first grasp the essence of linear equations. A linear equation is an algebraic expression where the highest power of the variable is 1. These equations, when graphed, form a straight line, hence the name 'linear'. The general form of a linear equation in one variable is ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The key to solving linear equations lies in isolating the variable on one side of the equation. This is achieved by performing the same operations on both sides, maintaining the equality. These operations include addition, subtraction, multiplication, and division. The goal is to simplify the equation until the variable stands alone, revealing its value. Linear equations are ubiquitous in various fields, from physics and engineering to economics and computer science. They model scenarios involving constant rates of change and provide a framework for predicting outcomes based on given conditions. Mastering the art of solving linear equations is therefore not just an academic exercise but a practical skill with widespread applications. As we dissect the equation 24p + 12 - 18p = 10 + 2p - 6, we will apply the fundamental principles of linear equation solving, demonstrating how to systematically isolate 'p' and uncover its numerical value.
Dissecting the Equation: 24p + 12 - 18p = 10 + 2p - 6
The given equation, 24p + 12 - 18p = 10 + 2p - 6, presents a typical linear equation structure. To solve for 'p', we need to streamline the equation by combining like terms and isolating the variable. The left-hand side of the equation contains two terms with 'p' (24p and -18p) and a constant term (12). Similarly, the right-hand side has a term with 'p' (2p) and constant terms (10 and -6). Our initial step involves simplifying both sides of the equation by combining these like terms. This process not only makes the equation more manageable but also brings us closer to isolating 'p'. By grouping the 'p' terms together and the constant terms together, we effectively consolidate the information within the equation. This simplification is a crucial step in solving any algebraic equation, as it reduces the complexity and allows us to focus on the essential components. As we proceed, we will meticulously apply algebraic operations to maintain the balance of the equation, ensuring that any transformation we make preserves the equality. The ultimate aim is to isolate 'p' on one side, thereby revealing its value and providing the solution to the equation. This systematic approach is a hallmark of mathematical problem-solving and underscores the importance of precision and clarity in algebraic manipulations.
Step-by-Step Solution
Let's embark on the journey of solving the equation 24p + 12 - 18p = 10 + 2p - 6 step by step. This meticulous approach ensures accuracy and clarity in the solution process.
1. Combine Like Terms on Both Sides
First, we combine the 'p' terms on the left-hand side: 24p - 18p = 6p. The left side now becomes 6p + 12. On the right-hand side, we combine the constant terms: 10 - 6 = 4. The right side simplifies to 2p + 4. The equation now looks like this: 6p + 12 = 2p + 4. This simplification is a crucial step in making the equation more manageable. By consolidating the terms, we reduce the clutter and create a clearer path towards isolating 'p'. This process of combining like terms is a fundamental technique in algebra, applicable to a wide range of equations. It allows us to group similar elements together, making the equation easier to analyze and solve. As we proceed, we will continue to apply these algebraic principles, ensuring that each step brings us closer to the solution.
2. Isolate the 'p' Terms
To isolate the 'p' terms, we subtract 2p from both sides of the equation: 6p + 12 - 2p = 2p + 4 - 2p. This simplifies to 4p + 12 = 4. Subtracting the same term from both sides maintains the equality, a core principle in solving equations. This step is a pivotal move in our quest to isolate 'p'. By eliminating the 'p' term from the right-hand side, we concentrate all the 'p' terms on the left-hand side. This strategic maneuver streamlines the equation, bringing us closer to our goal of determining the value of 'p'. The technique of subtracting the same term from both sides is a fundamental algebraic operation, applicable to a wide variety of equations. It allows us to manipulate the equation while preserving its balance, ensuring that the solution remains accurate. As we continue our journey, we will employ similar techniques to further isolate 'p', ultimately revealing its numerical value.
3. Isolate the Constant Terms
Next, we isolate the constant terms by subtracting 12 from both sides: 4p + 12 - 12 = 4 - 12. This simplifies to 4p = -8. This step is crucial in our quest to isolate 'p'. By eliminating the constant term from the left-hand side, we pave the way for the final step of solving for 'p'. The technique of subtracting the same value from both sides is a cornerstone of algebraic manipulation, ensuring that the equation remains balanced and the solution accurate. As we continue our journey, we will apply similar strategies to further isolate 'p', ultimately revealing its numerical value. The ability to manipulate equations in this manner is a fundamental skill in mathematics, applicable to a wide range of problems and disciplines.
4. Solve for 'p'
Finally, to solve for 'p', we divide both sides by 4: (4p) / 4 = -8 / 4. This gives us p = -2. This final step is the culmination of our efforts to isolate 'p'. By dividing both sides of the equation by the coefficient of 'p', we unveil its value, completing the solution process. The technique of dividing both sides by the same non-zero value is a fundamental algebraic operation, ensuring that the equation remains balanced and the solution accurate. With this step, we have successfully determined the value of 'p', which is -2. This solution satisfies the original equation, confirming the validity of our step-by-step approach.
The Value of p
Therefore, the value of p in the linear equation 24p + 12 - 18p = 10 + 2p - 6 is -2. This solution is the culmination of our step-by-step algebraic manipulation, where we meticulously simplified the equation and isolated the variable 'p'. The value p = -2 satisfies the original equation, meaning that when -2 is substituted for 'p', both sides of the equation are equal. This verification underscores the accuracy of our solution process. The journey to find 'p' involved combining like terms, isolating 'p' terms, and isolating constant terms, all while maintaining the balance of the equation through strategic algebraic operations. This process exemplifies the fundamental principles of solving linear equations, a crucial skill in mathematics and various scientific and engineering disciplines. The ability to solve for unknowns in linear relationships is essential for modeling and understanding real-world phenomena. As we conclude this exploration, we recognize the power and versatility of linear equations in providing solutions to a wide range of problems.
Conclusion
In conclusion, we have successfully navigated the linear equation 24p + 12 - 18p = 10 + 2p - 6 and determined the value of p to be -2. This journey underscored the importance of understanding and applying fundamental algebraic principles. From combining like terms to isolating the variable, each step played a crucial role in arriving at the solution. The process of solving linear equations is not merely an academic exercise; it is a fundamental skill with applications across various fields. Whether it's solving physics problems, optimizing engineering designs, or analyzing economic trends, the ability to manipulate and solve linear equations is invaluable. This exploration has reinforced the systematic approach required to tackle such problems, emphasizing the need for precision and clarity in algebraic manipulations. As we conclude, we recognize the significance of linear equations as a cornerstone of mathematical knowledge, paving the way for more advanced concepts and applications. The value of 'p' we found, -2, serves as a concrete example of how algebraic techniques can unravel the unknowns in mathematical expressions, providing us with solutions that have both theoretical and practical implications. This exercise not only solidifies our understanding of linear equations but also enhances our problem-solving capabilities in the broader realm of mathematics and beyond.
