Solving For X In 25x + (-63) = -13 A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving for the value of x in the equation 25x + (-63) = -13. This equation is a linear equation, a fundamental concept in algebra. Understanding how to solve such equations is crucial for various mathematical and real-world applications. We will break down the solution step by step, ensuring clarity and comprehension for readers of all levels. Our main keyword, solving for x, will be central to our discussion as we explore the techniques and principles involved. This article aims to provide a comprehensive guide, making the process accessible and straightforward.

Understanding Linear Equations

Before we dive into the specifics of the given equation, let's briefly discuss what linear equations are. 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. These equations are called "linear" because, when plotted on a graph, they form a straight line. The general form of a linear equation is ax + b = c, where x is the variable we aim to solve for, and a, b, and c are constants. The key to solving for x in a linear equation lies in isolating the variable on one side of the equation. This involves performing algebraic operations on both sides of the equation to maintain equality while gradually simplifying the equation until x is alone. These operations typically include addition, subtraction, multiplication, and division. Mastering linear equations is essential as they form the building blocks for more advanced mathematical concepts and are widely used in fields like physics, engineering, and economics. This understanding sets the stage for effectively tackling the equation at hand: 25x + (-63) = -13.

Step-by-Step Solution of 25x + (-63) = -13

To solve the equation 25x + (-63) = -13, we will follow a systematic approach, ensuring each step is clearly explained. The primary goal is to isolate x on one side of the equation. The first step in solving for x involves addressing the constant term on the same side as x. In this case, we have -63 added to 25x. To eliminate this -63, we perform the inverse operation, which is adding 63 to both sides of the equation. This maintains the balance of the equation, as whatever operation we perform on one side, we must also perform on the other. This gives us:

25x + (-63) + 63 = -13 + 63

Simplifying this, we get:

25x = 50

Now, we have 25 multiplied by x. To isolate x, we need to undo this multiplication. The inverse operation of multiplication is division. Therefore, we divide both sides of the equation by 25:

(25x) / 25 = 50 / 25

This simplifies to:

x = 2

Therefore, the value of x that makes the equation true is 2. This step-by-step process demonstrates the core principles of solving for x in linear equations.

Verification of the Solution

After solving for x, it's crucial to verify the solution to ensure accuracy. This involves substituting the obtained value of x back into the original equation to see if it holds true. In our case, we found that x = 2. So, we substitute 2 for x in the original equation:

25*(2) + (-63) = -13

Performing the multiplication, we get:

50 + (-63) = -13

Simplifying the addition, we have:

-13 = -13

Since the left side of the equation equals the right side, our solution x = 2 is correct. Verification is a vital step in problem-solving, especially in mathematics, as it confirms the validity of the solution and helps to avoid errors. This process not only ensures the correctness of the answer but also reinforces the understanding of the equation and the solving for x techniques used.

Common Mistakes to Avoid

When solving for x in linear equations, certain common mistakes can lead to incorrect solutions. Being aware of these pitfalls can significantly improve accuracy. One frequent error is not performing the same operation on both sides of the equation. Remember, the equation is a balance, and any operation done on one side must be mirrored on the other to maintain equality. Another common mistake is mishandling negative signs. It’s crucial to pay close attention to the signs of the numbers and apply the rules of arithmetic correctly. For instance, subtracting a negative number is equivalent to adding its positive counterpart. Another area of confusion can arise when dealing with the order of operations. It's essential to follow the correct order (PEMDAS/BODMAS) to ensure accurate simplification. Furthermore, failing to verify the solution is a significant oversight. As demonstrated earlier, verification is a simple yet powerful step that can catch errors. By being mindful of these common mistakes and practicing diligently, one can enhance their skills in solving for x and other algebraic problems.

Different Types of Linear Equations

While we have focused on a specific linear equation, it's important to recognize that linear equations come in various forms. Understanding these different types can broaden your problem-solving for x skills. The equation we solved, 25x + (-63) = -13, is a basic linear equation in the form ax + b = c. However, linear equations can also involve fractions, decimals, and parentheses. For instance, an equation like (1/2)x + 3 = 7 is a linear equation with a fraction. Solving such equations might require additional steps, such as multiplying both sides by the denominator to eliminate the fraction. Similarly, equations with parentheses, like 3(x - 2) = 9, need to be simplified by distributing the constant across the terms inside the parentheses before isolating x. Moreover, some linear equations might involve variables on both sides, such as 2x + 5 = x - 3. In these cases, the first step is usually to gather the variable terms on one side and the constants on the other. By recognizing the different forms linear equations can take and applying the appropriate algebraic techniques, you can confidently tackle a wide range of problems involving solving for x.

Real-World Applications of Linear Equations

The ability to solve for x in linear equations isn't just an academic exercise; it's a valuable skill with numerous real-world applications. Linear equations are used to model and solve problems in various fields, including physics, engineering, economics, and even everyday life. For example, in physics, linear equations can describe the motion of an object at a constant velocity. In engineering, they might be used to calculate the stress on a beam or the flow of current in a circuit. In economics, linear equations can model supply and demand curves, helping to determine market equilibrium. In personal finance, they can be used to calculate loan payments or investment returns. Even in simple everyday scenarios, linear equations can be applied. For instance, if you're trying to determine how many hours you need to work to earn a specific amount of money, you can set up a linear equation where x represents the number of hours worked. The versatility of linear equations makes solving for x a fundamental skill for problem-solving in a wide array of contexts. By mastering this skill, you equip yourself with a powerful tool for analyzing and addressing real-world challenges.

Conclusion

In this comprehensive guide, we have thoroughly explored the process of solving for x in the equation 25x + (-63) = -13. We started with an introduction to linear equations, then walked through the step-by-step solution, verified the answer, discussed common mistakes to avoid, examined different types of linear equations, and highlighted real-world applications. The key takeaway is that solving for x involves isolating the variable by performing inverse operations on both sides of the equation while maintaining balance. By understanding the principles and techniques discussed, you can confidently approach and solve a variety of linear equations. Remember, practice is key to mastering any mathematical skill. So, continue to solve different types of linear equations to further enhance your understanding and proficiency. The ability to solve for x is a fundamental skill that will serve you well in mathematics and beyond.