Solving -4.6p - 6.3p + 3.9 = -9.18 A Step-by-Step Guide
In this comprehensive guide, we will walk through the process of solving the equation -4.6p - 6.3p + 3.9 = -9.18. This is a common type of algebraic problem that involves combining like terms and isolating the variable. Understanding how to solve such equations is crucial for mastering algebra and related mathematical concepts. Our step-by-step approach will ensure that you not only arrive at the correct solution but also grasp the underlying principles. We will break down each step, providing clear explanations and insights, making this a valuable resource for students, educators, and anyone looking to enhance their problem-solving skills in mathematics.
The initial step in solving the equation -4.6p - 6.3p + 3.9 = -9.18 involves combining the like terms. Like terms are those that contain the same variable raised to the same power. In this equation, -4.6p and -6.3p are like terms because they both contain the variable p raised to the power of 1. To combine these terms, we simply add their coefficients. The coefficients are the numerical parts of the terms, which in this case are -4.6 and -6.3. Adding these together, we get -4.6 + (-6.3) = -10.9. This means that -4.6p - 6.3p simplifies to -10.9p. The equation now looks like this: -10.9p + 3.9 = -9.18. Combining like terms is a fundamental step in solving algebraic equations as it simplifies the equation, making it easier to work with and ultimately solve for the unknown variable. This process is not only applicable to linear equations but also extends to more complex equations involving polynomials and other algebraic expressions. By mastering this step, you build a solid foundation for tackling a wide range of mathematical problems.
After combining like terms, our equation stands as -10.9p + 3.9 = -9.18. The next crucial step is to isolate the term containing the variable p, which in this case is -10.9p. To do this, we need to eliminate the constant term on the same side of the equation, which is 3.9. We achieve this by performing the inverse operation. Since 3.9 is added to -10.9p, we subtract 3.9 from both sides of the equation. This maintains the equality of the equation, a fundamental principle in algebra. Subtracting 3.9 from both sides gives us: -10.9p + 3.9 - 3.9 = -9.18 - 3.9. On the left side, 3.9 - 3.9 cancels out, leaving us with -10.9p. On the right side, -9.18 - 3.9 equals -13.08. So, the equation now simplifies to -10.9p = -13.08. Isolating the variable term is a critical step because it brings us closer to finding the value of the variable itself. By isolating the variable term, we set the stage for the final step, which involves dividing to solve for the variable. This process of isolating terms is a cornerstone of algebraic manipulation and is used extensively in solving various types of equations.
Having isolated the variable term, our equation is now -10.9p = -13.08. To solve for p, we need to get p by itself. Since -10.9 is multiplied by p, we perform the inverse operation, which is division. We divide both sides of the equation by -10.9 to maintain the balance of the equation. This gives us: (-10.9p) / (-10.9) = (-13.08) / (-10.9). On the left side, -10.9 divided by -10.9 equals 1, leaving us with just p. On the right side, -13.08 divided by -10.9 results in approximately 1.2. Therefore, the solution to the equation is p = 1.2. This final step of dividing to solve for the variable is a common practice in algebra and is essential for finding the numerical value of the unknown. By performing this step accurately, we can determine the value of p that satisfies the original equation. The ability to solve for a variable is a fundamental skill in mathematics, applicable across various fields and disciplines.
To ensure the accuracy of our solution, it is always a good practice to verify it. We found that p = 1.2, so we substitute this value back into the original equation: -4.6p - 6.3p + 3.9 = -9.18. Substituting p = 1.2, we get: -4.6(1.2) - 6.3(1.2) + 3.9 = -9.18. Now, we perform the calculations. -4.6 multiplied by 1.2 is -5.52, and -6.3 multiplied by 1.2 is -7.56. So the equation becomes: -5.52 - 7.56 + 3.9 = -9.18. Adding -5.52 and -7.56 gives us -13.08. The equation now looks like: -13.08 + 3.9 = -9.18. Finally, adding -13.08 and 3.9 gives us -9.18, which matches the right side of the original equation. Since both sides of the equation are equal, our solution p = 1.2 is verified and correct. Verifying the solution is a crucial step in problem-solving as it confirms the accuracy of our calculations and reasoning. This practice helps to avoid errors and build confidence in our mathematical abilities.
In summary, we have successfully solved the equation -4.6p - 6.3p + 3.9 = -9.18 by following a step-by-step approach. We began by combining like terms, which simplified the equation. Then, we isolated the variable term by subtracting the constant from both sides. Finally, we solved for p by dividing both sides by the coefficient of p, which gave us the solution p = 1.2. We also verified our solution by substituting it back into the original equation, confirming its accuracy. This process demonstrates the importance of each step in solving algebraic equations and the need for careful attention to detail. Mastering these techniques is essential for success in algebra and further mathematical studies. By understanding the principles and methods outlined in this guide, you can confidently tackle similar equations and build a strong foundation in mathematical problem-solving.
Final Answer:
The solution to the equation is:
