Solving Linear Equations A Step-by-Step Guide To -3 + 8x - 5 = -8
In this article, we will delve into solving for the variable x in the linear equation -3 + 8x - 5 = -8. Linear equations are fundamental in mathematics and appear in various real-world applications, making it crucial to understand how to solve them effectively. This step-by-step guide will help you grasp the process, ensuring you can confidently tackle similar problems. We will cover simplifying the equation, isolating the variable, and arriving at the final solution. Whether you're a student learning algebra or someone looking to refresh your math skills, this article will provide a clear and comprehensive explanation.
Understanding Linear Equations
Before we jump into the solution, let's briefly discuss what linear equations are and why they are important. 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. The variable is raised to the power of one, and the equation, when graphed, forms a straight line. Linear equations are used extensively in various fields, including physics, engineering, economics, and computer science. They help model relationships between quantities and solve for unknown values. 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 we want to find.
The Importance of Solving Equations
Solving equations is a core skill in mathematics. It enables us to find the values that make an equation true, which is essential for problem-solving in various contexts. Whether it's determining the trajectory of a projectile, calculating the break-even point in a business, or designing a bridge, solving equations plays a vital role. Mastering this skill not only helps in academic pursuits but also in practical, everyday situations. In this article, we will focus on the specific techniques needed to solve the equation -3 + 8x - 5 = -8, providing you with the tools to approach similar problems confidently.
Breaking Down the Equation
Now, let's break down the equation -3 + 8x - 5 = -8. The goal is to isolate x on one side of the equation. This involves simplifying the equation by combining like terms and then using inverse operations to get x by itself. We will go through each step methodically, ensuring a clear understanding of the process. Remember, the key to solving equations is to maintain balance—whatever operation you perform on one side, you must also perform on the other side. This keeps the equation equivalent and leads us to the correct solution.
Step 1: Combine Like Terms
In the equation -3 + 8x - 5 = -8, we first need to combine the like terms. Like terms are terms that have the same variable raised to the same power. In this case, -3 and -5 are constants, so they are like terms. Combining them simplifies the equation:
-3 + 8x - 5 = -8
(-3 - 5) + 8x = -8
-8 + 8x = -8
Now our equation is simplified to -8 + 8x = -8. This step is crucial because it reduces the complexity of the equation, making it easier to isolate the variable x. By combining the constants, we have streamlined the equation and set the stage for the next steps in the solution process. Remember, always look for like terms to simplify equations before proceeding further.
Step 2: Isolate the Term with x
Next, we want to isolate the term with x. In the equation -8 + 8x = -8, the term with x is 8x. To isolate it, we need to eliminate the -8 on the left side of the equation. We can do this by adding 8 to both sides of the equation. This maintains the balance of the equation and moves us closer to isolating x:
-8 + 8x = -8
-8 + 8x + 8 = -8 + 8
8x = 0
Now we have 8x = 0. By adding 8 to both sides, we have successfully isolated the term containing x. This is a critical step in solving for x because it separates the variable term from the constants, making it easier to determine the value of x. The next step will involve dividing both sides by the coefficient of x to solve for the variable.
Step 3: Solve for x
Now that we have 8x = 0, the final step is to solve for x. To do this, we need to divide both sides of the equation by the coefficient of x, which is 8:
8x = 0
(8x) / 8 = 0 / 8
x = 0
Therefore, the solution to the equation -3 + 8x - 5 = -8 is x = 0. This is the value of x that makes the equation true. By dividing both sides by 8, we have successfully isolated x and found its value. This final step completes the solution process, giving us the answer to the equation.
The Solution
After following the steps to combine like terms, isolate the variable term, and solve for x, we have found that x = 0. This means that when x is replaced with 0 in the original equation, the equation holds true. To confirm our solution, we can substitute x = 0 back into the original equation and verify that both sides are equal.
Verification
Let's verify our solution by substituting x = 0 into the original equation:
-3 + 8x - 5 = -8
-3 + 8(0) - 5 = -8
-3 + 0 - 5 = -8
-8 = -8
Since both sides of the equation are equal, our solution x = 0 is correct. Verification is an important step in solving equations because it ensures that the value we found is indeed the correct solution. By substituting the value back into the original equation, we can confirm that our solution satisfies the equation.
Conclusion
In conclusion, we have successfully solved the equation -3 + 8x - 5 = -8 and found that x = 0. This process involved combining like terms, isolating the term with x, and dividing both sides by the coefficient of x. We also verified our solution by substituting it back into the original equation. Solving linear equations is a fundamental skill in mathematics, and by understanding these steps, you can confidently tackle similar problems. Remember to always simplify the equation, isolate the variable, and verify your solution to ensure accuracy. With practice, solving linear equations will become second nature, and you will be well-equipped to handle more complex mathematical challenges.
The correct answer is D. 0.
