Solving For X In The Equation -8 = (-10 + X) / 4

Understanding the Problem

In this article, we will delve into solving a fundamental algebraic equation. Our primary goal is to find the value of the variable x that satisfies the equation -8 = (-10 + x) / 4. This type of problem is a cornerstone of algebra and is essential for understanding more complex mathematical concepts. We will explore the step-by-step process to isolate x and determine its value, providing a clear and concise explanation along the way. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide will help you master the techniques needed to solve similar equations. Let's embark on this mathematical journey together and unravel the mystery of x.

Algebraic equations are the backbone of mathematical problem-solving, and understanding how to solve them is crucial for various fields, including science, engineering, and finance. The equation -8 = (-10 + x) / 4 is a linear equation, which means it involves a single variable (x) raised to the power of one. Solving such equations involves isolating the variable on one side of the equation to determine its value. This process often requires performing a series of operations, such as addition, subtraction, multiplication, and division, while maintaining the equality of both sides. In this case, we will start by eliminating the fraction and then isolate x using basic arithmetic operations. By the end of this discussion, you will have a clear understanding of how to approach and solve this type of equation, equipping you with a valuable skill for tackling more advanced mathematical problems. Remember, the key to success in algebra is practice, so feel free to try out similar equations and reinforce your understanding.

The process of solving for x in the equation -8 = (-10 + x) / 4 is a journey through the fundamental principles of algebra. We begin by acknowledging the structure of the equation, noting that x is entangled within a fraction. Our initial strategy involves dismantling this fraction to liberate x. This is achieved by multiplying both sides of the equation by the denominator, which in this case is 4. This pivotal step preserves the equation's balance while simplifying its appearance. Next, we shift our focus to isolating x entirely. This requires counteracting the constant term that accompanies x, which is -10. We employ the inverse operation, adding 10 to both sides of the equation, ensuring that the scales remain balanced. As we progress through these steps, we witness the gradual unveiling of x's true value. Each operation is a deliberate move, guided by the principles of algebraic manipulation. The final step reveals the solution, the precise numerical value that satisfies the equation. This methodical approach not only solves the immediate problem but also cultivates a deeper understanding of algebraic problem-solving, empowering us to tackle more intricate equations with confidence. The beauty of algebra lies in its systematic nature, where each step builds upon the previous one, leading us to the ultimate solution.

Step-by-Step Solution

1. Eliminate the Fraction

To get rid of the fraction, we multiply both sides of the equation by 4:

-8 = (-10 + x) / 4
-8 * 4 = ((-10 + x) / 4) * 4
-32 = -10 + x

Multiplying both sides by 4 is a crucial first step in isolating x. This operation effectively cancels out the denominator on the right side of the equation, simplifying the expression and making it easier to work with. By performing the multiplication, we transform the equation from one involving a fraction to a more straightforward linear equation. This step demonstrates the fundamental principle of maintaining equality in an equation: whatever operation is performed on one side must also be performed on the other. In this case, multiplying both sides by 4 ensures that the equation remains balanced and the value of x is not altered. This technique is widely used in algebra and is essential for solving equations involving fractions. It allows us to clear the fractions and proceed with isolating the variable using simpler arithmetic operations. The result, -32 = -10 + x, is now much easier to manipulate, bringing us closer to finding the value of x.

2. Isolate x

Now, to isolate x, we add 10 to both sides of the equation:

-32 = -10 + x
-32 + 10 = -10 + x + 10
-22 = x

Isolating x is the key to solving the equation. In this step, we focus on removing the constant term (-10) that is added to x on the right side of the equation. To do this, we apply the inverse operation, which is addition. By adding 10 to both sides of the equation, we maintain the equality while effectively canceling out the -10 on the right side. This leaves x isolated on one side, allowing us to directly determine its value. The principle of adding the same value to both sides is fundamental in algebra, ensuring that the equation remains balanced and the solution remains accurate. This step is a clear demonstration of how inverse operations are used to simplify equations and isolate variables. The resulting equation, -22 = x, directly reveals the value of x, which is -22. This is the final step in solving the equation, providing us with the solution we were seeking.

3. Solution

Therefore, the solution to the equation is:

x = -22

After performing the necessary algebraic manipulations, we have arrived at the solution: x = -22. This value is the unique solution to the equation -8 = (-10 + x) / 4, meaning that when x is replaced with -22 in the original equation, the equation holds true. This solution represents the point where the left-hand side of the equation equals the right-hand side, satisfying the mathematical relationship defined by the equation. The process of solving for x has involved eliminating the fraction and isolating the variable, demonstrating the core principles of algebraic problem-solving. The result, x = -22, is a concise and definitive answer that can be readily applied in any context where this equation arises. This process not only provides the solution but also reinforces the understanding of how algebraic equations are solved, building confidence in tackling more complex problems in the future. The ability to solve for variables is a fundamental skill in mathematics and is essential for various applications in science, engineering, and other fields. Therefore, mastering these techniques is crucial for anyone pursuing further studies or careers in these areas.

Verification

To verify our solution, we substitute x = -22 back into the original equation:

-8 = (-10 + x) / 4
-8 = (-10 + (-22)) / 4
-8 = (-10 - 22) / 4
-8 = (-32) / 4
-8 = -8

The equation holds true, so our solution is correct.

Verifying the solution is a critical step in the problem-solving process. It ensures that the value we have found for x is indeed correct and satisfies the original equation. By substituting x = -22 back into the equation -8 = (-10 + x) / 4, we can confirm its validity. This process involves replacing x with -22 and then simplifying both sides of the equation to see if they are equal. In this case, substituting -22 for x gives us -8 = (-10 + (-22)) / 4. Simplifying the right-hand side, we get -8 = (-32) / 4, which further simplifies to -8 = -8. Since both sides of the equation are equal, this confirms that our solution, x = -22, is correct. This verification step not only provides assurance that we have solved the problem accurately but also reinforces our understanding of the equation and its solution. It highlights the importance of checking our work to avoid errors and ensures that we can confidently use the solution in further calculations or applications. This practice is essential for developing strong problem-solving skills and is a hallmark of rigorous mathematical thinking.

Conclusion

In conclusion, we have successfully solved the equation -8 = (-10 + x) / 4 and found that x = -22. We accomplished this by first eliminating the fraction through multiplication and then isolating x by using inverse operations. Finally, we verified our solution by substituting it back into the original equation. This step-by-step process demonstrates the fundamental principles of algebra and provides a clear method for solving similar equations. Remember, practice is key to mastering these skills, so continue to explore different equations and apply these techniques to strengthen your understanding.

The journey of solving the equation -8 = (-10 + x) / 4 has been a valuable exercise in applying algebraic principles. We began by identifying the structure of the equation and recognizing the need to isolate x. The initial step of multiplying both sides by 4 was crucial in eliminating the fraction and simplifying the equation. This allowed us to move forward with the process of isolating x, which we achieved by adding 10 to both sides. The resulting solution, x = -22, represents the value that satisfies the equation, making the left-hand side equal to the right-hand side. The verification step further solidified our understanding by confirming that this value indeed holds true when substituted back into the original equation. This comprehensive approach not only solves the immediate problem but also reinforces the underlying concepts of algebra, such as inverse operations and maintaining equality. By mastering these techniques, we gain the confidence to tackle more complex equations and apply these skills in various mathematical and real-world contexts. The ability to solve equations is a fundamental skill that opens doors to further exploration in mathematics and its applications.