Solving -15x - 50 = 25 A Step-by-Step Guide

Introduction: Mastering Linear Equations

In the realm of mathematics, linear equations form the bedrock of algebraic problem-solving. These equations, characterized by a single variable raised to the first power, appear ubiquitously in various scientific and engineering disciplines. The ability to solve linear equations efficiently and accurately is thus a fundamental skill for students and professionals alike. This article delves into the process of solving a specific linear equation, -15x - 50 = 25, providing a step-by-step guide and elucidating the underlying principles. We will not only arrive at the correct solution but also explore the rationale behind each step, fostering a deeper understanding of the concepts involved. The equation we will tackle, -15x - 50 = 25, is a classic example of a linear equation in one variable. Our objective is to isolate the variable 'x' on one side of the equation, thereby determining its value. This involves applying a series of algebraic manipulations, adhering to the fundamental principle that any operation performed on one side of the equation must also be performed on the other side to maintain equality. Let's embark on this journey of mathematical exploration and unravel the value of 'x'.

Step 1: Isolating the Variable Term

The primary goal in solving any linear equation is to isolate the term containing the variable. In our equation, -15x - 50 = 25, the term with the variable is -15x. To isolate this term, we need to eliminate the constant term, which is -50. The principle of inverse operations comes into play here. Since we have a subtraction of 50, we perform the inverse operation, which is addition. We add 50 to both sides of the equation. This crucial step ensures that the equation remains balanced. Adding 50 to both sides, we get:

-15x - 50 + 50 = 25 + 50

Simplifying this, the -50 and +50 on the left side cancel each other out, leaving us with:

-15x = 75

Now, the variable term -15x is isolated on the left side of the equation. This is a significant milestone in our journey to solve for 'x'. By understanding the concept of inverse operations and applying it judiciously, we have successfully taken the first step towards finding the solution. The next step will involve isolating 'x' itself, which we will address in the subsequent section. Remember, the key to solving linear equations lies in systematically applying algebraic operations while maintaining the balance of the equation.

Step 2: Solving for x

Having isolated the variable term, -15x = 75, the next logical step is to isolate 'x' itself. Currently, 'x' is being multiplied by -15. To undo this multiplication, we employ the inverse operation, which is division. We divide both sides of the equation by -15. This step is crucial to ensure that we maintain the equality of the equation. Dividing both sides by -15, we get:

-15x / -15 = 75 / -15

On the left side, -15 divided by -15 equals 1, effectively isolating 'x'. On the right side, 75 divided by -15 equals -5. Thus, the equation simplifies to:

x = -5

We have now successfully solved for 'x'. The value of 'x' that satisfies the original equation -15x - 50 = 25 is -5. This solution represents the point where the equation holds true. The process of dividing both sides by the coefficient of 'x' is a fundamental technique in solving linear equations. It allows us to isolate the variable and determine its value. In this case, dividing by -15 was the key to unlocking the solution. It's important to remember that the sign of the coefficient plays a crucial role in the division process. A negative coefficient, as in this case, results in a sign change during the division. With 'x' now isolated, we have the solution. However, it's always a good practice to verify the solution, which we will do in the next section.

Step 3: Verifying the Solution

After obtaining a solution, it's essential to verify its correctness. This step provides a safeguard against potential errors in the algebraic manipulations. To verify our solution, x = -5, we substitute this value back into the original equation, -15x - 50 = 25. If the left side of the equation equals the right side after the substitution, our solution is correct.

Substituting x = -5 into the equation, we get:

-15(-5) - 50 = 25

First, we perform the multiplication: -15 multiplied by -5 equals 75. Remember that the product of two negative numbers is positive. So, the equation becomes:

75 - 50 = 25

Next, we perform the subtraction: 75 minus 50 equals 25. Thus, the equation simplifies to:

25 = 25

Since the left side of the equation is equal to the right side, our solution, x = -5, is indeed correct. This verification step not only confirms the accuracy of our solution but also reinforces our understanding of the equation. By substituting the solution back into the original equation, we are essentially checking if the value of 'x' we found satisfies the equation's conditions. This process is a cornerstone of problem-solving in mathematics and other scientific disciplines. With the solution verified, we can confidently state that we have successfully solved the linear equation.

Conclusion: The Power of Algebraic Manipulation

In this comprehensive exploration, we have successfully solved the linear equation -15x - 50 = 25. Through a step-by-step approach, we first isolated the variable term by adding 50 to both sides of the equation. Then, we isolated 'x' by dividing both sides by -15. This meticulous process led us to the solution, x = -5. Subsequently, we verified our solution by substituting it back into the original equation, confirming its accuracy. This exercise highlights the power of algebraic manipulation in solving mathematical problems. Linear equations, like the one we tackled, are fundamental in various fields, including physics, engineering, economics, and computer science. The ability to solve these equations efficiently and accurately is a valuable skill that can be applied to a wide range of real-world scenarios. The key to mastering linear equations lies in understanding the underlying principles and practicing consistently. By grasping the concepts of inverse operations and maintaining the balance of the equation, you can confidently tackle any linear equation that comes your way. Remember, mathematics is not just about finding the right answer; it's about understanding the process and the reasoning behind each step. This article aimed to provide not only the solution but also a clear explanation of the process, empowering you to approach similar problems with confidence and competence. The journey of mathematical exploration is continuous, and each solved problem contributes to a deeper understanding of the subject. So, keep practicing, keep exploring, and keep solving!