Solving For P In The Equation 24p + 12 - 18p = 10 + 2p - 6
Introduction
In this article, we will delve into the process of solving a linear equation for the variable p. Linear equations are fundamental to algebra and are used extensively in various fields, including mathematics, physics, engineering, and economics. Understanding how to solve them is a crucial skill for anyone working with quantitative data or mathematical models. The specific equation we will address is 24p + 12 - 18p = 10 + 2p - 6. This equation might seem complex at first glance, but by following a systematic approach, we can simplify it and isolate p to find its value. Our discussion will cover each step in detail, ensuring clarity and comprehension. This includes combining like terms, moving variables and constants to appropriate sides of the equation, and finally, solving for p. This step-by-step guide not only provides the solution but also reinforces the underlying principles of algebraic manipulation. Mastering these principles allows for the solution of a wide array of linear equations, making it an invaluable skill for further mathematical studies and practical applications. Whether you're a student learning algebra for the first time or someone seeking to refresh your skills, this article aims to provide a comprehensive and understandable explanation. By the end of this guide, you'll be equipped with the knowledge and confidence to tackle similar problems with ease. Remember, the key to success in algebra lies in understanding the basic rules and applying them methodically. So, let's begin our journey to solve for p and unlock the power of linear equations.
Understanding Linear Equations
Linear equations, at their core, represent relationships where the highest power of the variable is 1. This characteristic distinguishes them from quadratic equations (where the highest power is 2), cubic equations (where the highest power is 3), and so on. The general form of a linear equation in one variable, such as p, can be expressed as ap + b = c, where a, b, and c are constants, and a is not equal to zero. The goal when solving a linear equation is to isolate the variable on one side of the equation to determine its value. This isolation is achieved by performing the same operations on both sides of the equation, ensuring that the equality is maintained. These operations typically involve addition, subtraction, multiplication, and division. Consider, for instance, the equation 3p + 5 = 14. To solve for p, we first subtract 5 from both sides, resulting in 3p = 9. Then, we divide both sides by 3 to get p = 3. This simple example illustrates the fundamental principles involved in solving linear equations. Now, when dealing with more complex linear equations like 24p + 12 - 18p = 10 + 2p - 6, the same principles apply, but the equation requires additional steps to simplify before isolating the variable. These steps often involve combining like terms and rearranging the equation to group the variable terms on one side and the constant terms on the other. Understanding these foundational concepts is crucial for tackling more advanced algebraic problems. Linear equations form the basis for many mathematical models used in various fields, making their mastery essential for students and professionals alike. By grasping the mechanics of solving linear equations, individuals gain a valuable tool for problem-solving and critical thinking, which can be applied in numerous real-world scenarios.
Step 1: Simplify Both Sides of the Equation
The initial step in solving the linear equation 24p + 12 - 18p = 10 + 2p - 6 is to simplify both sides of the equation separately. This involves combining like terms, which are terms that have the same variable raised to the same power (in this case, just p to the power of 1) or are constants (numbers without a variable). On the left side of the equation, we have two terms with p: 24p and -18p. Combining these terms involves adding their coefficients: 24 + (-18) = 6. So, 24p - 18p simplifies to 6p. The left side also has a constant term, which is 12. Therefore, the simplified form of the left side of the equation is 6p + 12. On the right side of the equation, we have the terms 10, 2p, and -6. There is only one term with p, which is 2p. The constant terms are 10 and -6. Combining these constant terms involves subtracting 6 from 10, which gives us 4. So, 10 - 6 simplifies to 4. The simplified form of the right side of the equation is 2p + 4. Now, after simplifying both sides, the original equation 24p + 12 - 18p = 10 + 2p - 6 has been reduced to a simpler form: 6p + 12 = 2p + 4. This simplified equation is much easier to work with and represents the same relationship as the original equation. Simplifying both sides before proceeding further is a crucial step because it reduces the complexity of the equation and makes subsequent steps less prone to errors. By combining like terms, we have effectively condensed the equation into a more manageable form, setting the stage for the next steps in solving for p. This methodical approach to problem-solving is a key skill in algebra and mathematics in general.
Step 2: Move the Variable Terms to One Side
After simplifying both sides of the equation, the next crucial step in solving for p is to consolidate all variable terms on one side of the equation. In our simplified equation, 6p + 12 = 2p + 4, we have p terms on both the left and right sides. To isolate p, we need to move one of these terms to the other side. A common strategy is to move the term with the smaller coefficient to avoid dealing with negative coefficients, but either choice will lead to the correct answer if done properly. In this case, 2p has a smaller coefficient than 6p, so we will move 2p from the right side to the left side. To do this, we subtract 2p from both sides of the equation. This maintains the equality because we are performing the same operation on both sides. Subtracting 2p from the left side, 6p + 12, gives us 6p - 2p + 12, which simplifies to 4p + 12. Subtracting 2p from the right side, 2p + 4, gives us 2p - 2p + 4, which simplifies to 4. Now, our equation looks like this: 4p + 12 = 4. By moving the variable terms to the left side, we have successfully grouped the p terms together. This step brings us closer to isolating p and finding its value. The key to this process is to perform the same operation on both sides of the equation to maintain balance. This ensures that the equation remains equivalent to the original equation throughout the solving process. This step is a fundamental algebraic manipulation technique that is widely used in solving various types of equations. By understanding and mastering this technique, you can confidently approach more complex equations and algebraic problems.
Step 3: Move the Constant Terms to the Other Side
Following the consolidation of variable terms on one side, the next logical step in solving for p is to move all the constant terms to the opposite side. In our current equation, 4p + 12 = 4, we have the variable term 4p on the left side and the constant term 12 also on the left side. Our goal is to isolate the variable term, so we need to move the constant term 12 to the right side of the equation. To do this, we perform the inverse operation of addition, which is subtraction. We subtract 12 from both sides of the equation to maintain the equality. Subtracting 12 from the left side, 4p + 12, gives us 4p + 12 - 12, which simplifies to 4p. Subtracting 12 from the right side, 4, gives us 4 - 12, which equals -8. Now, our equation looks like this: 4p = -8. By moving the constant term to the right side, we have successfully isolated the variable term on the left side. This is a crucial step towards finding the value of p. The process of moving constant terms involves using inverse operations to cancel out the term on one side of the equation and reintroduce it on the other side. This technique is a fundamental aspect of solving equations and is based on the principle of maintaining balance by performing the same operation on both sides. By understanding and applying this technique, you can systematically isolate variables and solve for their values in various algebraic equations. This step-by-step approach is essential for accurate and efficient problem-solving in mathematics and related fields.
Step 4: Solve for p
Having isolated the variable term, the final step in solving for p is to divide both sides of the equation by the coefficient of p. In our current equation, 4p = -8, the coefficient of p is 4. To isolate p, we need to undo the multiplication by 4. We do this by dividing both sides of the equation by 4. Dividing the left side, 4p, by 4 gives us (4p)/4, which simplifies to p. Dividing the right side, -8, by 4 gives us -8/4, which equals -2. Therefore, we have found that p = -2. This solution means that if we substitute -2 for p in the original equation, 24p + 12 - 18p = 10 + 2p - 6, both sides of the equation will be equal. We can verify this by plugging in -2 for p in the original equation: Left side: 24(-2) + 12 - 18(-2) = -48 + 12 + 36 = 0 Right side: 10 + 2(-2) - 6 = 10 - 4 - 6 = 0 Since both sides equal 0, our solution p = -2 is correct. This final step of dividing by the coefficient of the variable is a fundamental technique in solving equations. It allows us to isolate the variable and determine its value. The process involves using the inverse operation of multiplication, which is division, to cancel out the coefficient. This step, combined with the previous steps of simplifying, moving variable terms, and moving constant terms, provides a comprehensive approach to solving linear equations. By mastering these techniques, you can confidently solve a wide range of algebraic problems and apply these skills in various mathematical and real-world contexts.
Conclusion
In conclusion, solving the linear equation 24p + 12 - 18p = 10 + 2p - 6 involves a systematic approach that includes simplifying both sides, moving variable terms to one side, moving constant terms to the other side, and finally, solving for p. We began by simplifying both sides of the equation, combining like terms to reduce complexity. This resulted in the simplified equation 6p + 12 = 2p + 4. Next, we moved the variable terms to one side by subtracting 2p from both sides, resulting in 4p + 12 = 4. Then, we moved the constant terms to the other side by subtracting 12 from both sides, which gave us 4p = -8. Finally, we solved for p by dividing both sides by 4, which led us to the solution p = -2. We also verified this solution by substituting -2 for p in the original equation and confirming that both sides were equal. This step-by-step process illustrates the fundamental principles of solving linear equations, which are crucial for algebra and various other mathematical applications. Each step involves using inverse operations to isolate the variable while maintaining the equality of the equation. This approach not only provides the solution but also reinforces the underlying algebraic concepts. Mastering these techniques enables you to confidently tackle more complex equations and problems. The ability to solve linear equations is a valuable skill that extends beyond mathematics, finding applications in fields such as physics, engineering, economics, and computer science. By understanding and practicing these methods, you can enhance your problem-solving abilities and analytical thinking, which are essential for success in many academic and professional pursuits. Therefore, the process of solving linear equations, as demonstrated in this article, is not just about finding a numerical answer but also about developing a logical and methodical approach to problem-solving.
