Solving -8x + 5 = -10 - 25 A Step-by-Step Guide

Introduction

In the realm of mathematics, solving linear equations is a fundamental skill. It forms the bedrock for more advanced topics in algebra and calculus. Linear equations, characterized by a variable raised to the power of one, appear frequently in various real-world scenarios, from calculating finances to determining distances and rates. This article provides a detailed, step-by-step guide on how to solve the linear equation -8x + 5 = -10 - 25. By understanding the process involved, you'll gain confidence in tackling similar problems and build a strong foundation for your mathematical journey. We will break down each step, explaining the logic behind the operations and ensuring clarity for learners of all levels. Whether you are a student struggling with algebra or someone looking to refresh your mathematical skills, this guide will offer valuable insights and practical techniques.

Understanding the Equation

Before diving into the solution, let’s first understand the structure of the linear equation -8x + 5 = -10 - 25. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. In this case, the variable is 'x'. The equation consists of two sides, the left-hand side (LHS) which is -8x + 5, and the right-hand side (RHS) which is -10 - 25. The goal of solving an equation is to isolate the variable on one side, thereby determining the value of 'x' that makes the equation true. To achieve this, we employ various algebraic operations, ensuring that we maintain the balance of the equation. This means that any operation performed on one side must also be performed on the other side. Understanding this fundamental principle is crucial for accurately solving equations. The terms in the equation each play a role: -8x represents a variable term, 5 is a constant term on the LHS, and -10 and -25 are constants on the RHS. By manipulating these terms correctly, we can systematically arrive at the solution.

Step 1: Simplify Both Sides of the Equation

The first step in solving the equation -8x + 5 = -10 - 25 is to simplify both sides independently. This involves combining any like terms present on either side. On the left-hand side (LHS), we have -8x + 5. There are no like terms to combine here, as -8x is a variable term and 5 is a constant term. Thus, the LHS remains as -8x + 5. On the right-hand side (RHS), we have -10 - 25. These are both constant terms and can be combined. Subtracting 25 from -10 gives us -35. So, the RHS simplifies to -35. Now, our equation looks like this: -8x + 5 = -35. This simplification step is crucial as it reduces the complexity of the equation, making it easier to work with in subsequent steps. By performing basic arithmetic operations, we streamline the equation, which helps in isolating the variable. Simplifying both sides is a common first step in solving any algebraic equation, and it is important to perform it accurately to avoid errors in the final solution.

Step 2: Isolate the Variable Term

After simplifying both sides of the equation -8x + 5 = -35, the next step is to isolate the variable term, which in this case is -8x. To isolate the variable term, we need to eliminate any constant terms that are on the same side of the equation. In this instance, we have the constant +5 on the left-hand side. To eliminate +5, we perform the inverse operation, which is subtraction. We subtract 5 from both sides of the equation. This maintains the balance of the equation, ensuring that we are performing the same operation on both sides. Subtracting 5 from the LHS gives us: -8x + 5 - 5, which simplifies to -8x. Subtracting 5 from the RHS gives us: -35 - 5, which simplifies to -40. Our equation now looks like this: -8x = -40. By subtracting 5 from both sides, we have successfully isolated the variable term -8x on the left-hand side. This step is a critical part of the process, as it brings us closer to solving for the variable 'x'.

Step 3: Solve for the Variable

With the equation now in the form -8x = -40, the final step is to solve for the variable 'x'. To do this, we need to isolate 'x' completely. Currently, 'x' is being multiplied by -8. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by -8. Dividing both sides by the same number maintains the equation's balance. Dividing the LHS -8x by -8 gives us 'x', as -8 divided by -8 is 1. Dividing the RHS -40 by -8 gives us 5, as -40 divided by -8 is 5. Therefore, our solution is x = 5. This means that the value of 'x' that makes the original equation -8x + 5 = -10 - 25 true is 5. By dividing both sides by -8, we have successfully solved for 'x', arriving at the final answer. This step demonstrates the power of using inverse operations to isolate and solve for a variable in a linear equation.

Step 4: Check Your Solution

After solving for the variable, it's crucial to check your solution to ensure accuracy. To verify the solution x = 5 for the equation -8x + 5 = -10 - 25, we substitute the value of 'x' back into the original equation. This process involves replacing every instance of 'x' in the equation with the value 5. Substituting x = 5 into the LHS -8x + 5 gives us: -8(5) + 5. Multiplying -8 by 5 gives us -40, so the LHS becomes -40 + 5. Adding 5 to -40 gives us -35. Thus, the LHS simplifies to -35. The RHS of the equation is -10 - 25, which we previously simplified to -35. Now, we compare the simplified LHS and RHS: -35 = -35. Since both sides are equal, our solution x = 5 is correct. Checking your solution is an essential practice in mathematics as it helps to identify any errors made during the solving process. By substituting the solution back into the original equation, we confirm that the value of the variable indeed satisfies the equation. This step provides confidence in the correctness of the solution and reinforces the understanding of the equation-solving process.

Conclusion

In conclusion, solving the linear equation -8x + 5 = -10 - 25 involves a series of methodical steps: simplifying both sides, isolating the variable term, solving for the variable, and checking the solution. We began by simplifying the RHS to -35, resulting in the equation -8x + 5 = -35. Next, we isolated the variable term by subtracting 5 from both sides, yielding -8x = -40. We then solved for 'x' by dividing both sides by -8, which gave us the solution x = 5. Finally, we verified our solution by substituting x = 5 back into the original equation, confirming that both sides equaled -35. This step-by-step guide demonstrates the importance of understanding each operation and its effect on the equation. Mastering these techniques is crucial for success in algebra and beyond. By following a structured approach and practicing regularly, you can confidently solve linear equations and build a solid foundation in mathematics. The ability to solve linear equations is not only essential in academic settings but also in various real-life applications, making it a valuable skill to acquire.