Solving For X In The Equation -5x + X = -40

Introduction

In the realm of algebra, a fundamental skill is the ability to solve equations. This involves manipulating equations to isolate the unknown variable, thereby determining the value that satisfies the given relationship. This article delves into the step-by-step process of solving the equation $-5x + x = -40$, providing a clear and concise explanation suitable for students and anyone seeking to enhance their algebraic proficiency. We will begin by simplifying the equation, combining like terms, and then isolating the variable x to find its value. This process underscores the core principles of equation solving, emphasizing the importance of maintaining balance and performing operations on both sides to preserve equality. The practical application of these skills extends beyond the classroom, finding relevance in various fields such as engineering, finance, and computer science, where mathematical modeling and problem-solving are paramount.

Understanding the Equation

Before diving into the solution, it's crucial to understand the equation itself. The equation $-5x + x = -40$ represents a linear relationship where x is the unknown variable. The left-hand side of the equation, $-5x + x$, involves two terms containing the variable x, while the right-hand side, $-40$, is a constant. Our goal is to find the value of x that makes the left-hand side equal to the right-hand side. To achieve this, we will employ algebraic manipulations based on the properties of equality. These properties allow us to perform operations on both sides of the equation without changing its solution. For instance, adding or subtracting the same value from both sides, or multiplying or dividing both sides by the same non-zero value, maintains the equation's balance. Understanding these fundamental principles is key to successfully navigating the solution process and developing a strong foundation in algebra. The equation at hand serves as a valuable example for illustrating these concepts, offering a clear path to understanding how to isolate variables and solve for their values.

Step 1: Combining Like Terms

The first step in solving the equation $-5x + x = -40$ is to combine like terms. On the left-hand side, we have two terms containing the variable x: $-5x$ and $+x$. These are like terms because they both involve the same variable raised to the same power (in this case, x to the power of 1). To combine them, we simply add their coefficients. The coefficient of $-5x$ is -5, and the coefficient of $+x$ is 1 (since x is the same as $1x$). Adding the coefficients, we get $-5 + 1 = -4$. Therefore, $-5x + x$ simplifies to $-4x$. The equation now becomes $-4x = -40$. This simplification is a crucial step as it reduces the complexity of the equation, making it easier to isolate the variable. By combining like terms, we've essentially condensed the expression, bringing us closer to our goal of finding the value of x. This process highlights the importance of recognizing and grouping similar terms in algebraic expressions, a skill that is fundamental to solving a wide range of equations and mathematical problems. The ability to combine like terms efficiently streamlines the solution process and minimizes the chances of errors.

Step 2: Isolating the Variable

Having simplified the equation to $-4x = -40$, the next step is to isolate the variable x. This means getting x by itself on one side of the equation. Currently, x is being multiplied by -4. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by -4. This is crucial to maintain the balance of the equation; whatever operation we perform on one side, we must also perform on the other. Dividing both sides by -4, we get: $(-4x) / -4 = -40 / -4$. On the left-hand side, the -4 in the numerator and the -4 in the denominator cancel each other out, leaving us with just x. On the right-hand side, we have -40 divided by -4. When dividing two negative numbers, the result is positive. So, $-40 / -4 = 10$. Therefore, the equation simplifies to $x = 10$. This isolation step is the heart of equation solving, where we strategically use inverse operations to unravel the equation and reveal the value of the unknown variable. The careful application of division in this case has successfully isolated x, leading us to the solution.

Step 3: Solution and Verification

After isolating the variable, we have found the solution to the equation: $x = 10$. This means that the value of x that satisfies the original equation $-5x + x = -40$ is 10. However, to ensure the accuracy of our solution, it's always a good practice to verify it. Verification involves substituting the obtained value of x back into the original equation to see if it holds true. Substituting $x = 10$ into the original equation, we get: $-5(10) + 10 = -40$. Simplifying the left-hand side, we have $-50 + 10 = -40$. Indeed, $-40 = -40$, which confirms that our solution is correct. This verification step is a valuable safeguard against errors, providing confidence in the accuracy of the solution. By substituting the calculated value back into the original equation, we ensure that the equation's balance is maintained. This practice reinforces the understanding of equation solving and the importance of checking one's work. In this case, the verification confirms that $x = 10$ is the correct solution, demonstrating the successful application of algebraic principles.

Conclusion

In conclusion, we have successfully found the value of the variable x that satisfies the equation $-5x + x = -40$. Through a series of algebraic steps, we simplified the equation, combined like terms, isolated the variable, and verified the solution. The process involved combining the terms $-5x$ and $x$ to get $-4x$, then dividing both sides of the equation $-4x = -40$ by -4 to isolate x, resulting in $x = 10$. The solution was then verified by substituting $x = 10$ back into the original equation, confirming its accuracy. This exercise underscores the fundamental principles of equation solving in algebra, highlighting the importance of maintaining balance, applying inverse operations, and verifying solutions. These skills are essential for success in mathematics and have broad applications in various fields. The ability to solve equations is a cornerstone of mathematical literacy, enabling individuals to tackle a wide range of problems and make informed decisions based on quantitative reasoning. This article provides a clear and concise guide to solving linear equations, empowering readers to confidently approach similar problems and further develop their algebraic skills.