Solving -1/4(2x+8)=-6 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. Linear equations, in particular, form the bedrock of many mathematical concepts and real-world applications. This article delves into the process of solving a specific linear equation, $-\frac{1}{4}(2x+8)=-6$, providing a detailed, step-by-step guide to unraveling the unknown variable. By understanding the underlying principles and techniques, you'll gain the confidence to tackle similar equations with ease. Let's embark on this mathematical journey together.

Understanding the Equation

At its core, the equation $-\frac{1}{4}(2x+8)=-6$ is a statement of equality between two expressions. Our goal is to isolate the variable x on one side of the equation, thereby determining its value. To achieve this, we'll employ a series of algebraic manipulations, ensuring that we maintain the balance of the equation at each step. It is very important to understand the structure of the given equation. We can see that the equation involves a fractional coefficient multiplied by a term in parentheses, which is equal to a constant. The variable x is hidden inside the parentheses, so our first step will involve distributing the coefficient.

Step 1: Distribute the Coefficient

The initial step in solving the equation is to eliminate the parentheses by distributing the coefficient $-\frac{1}{4}$ across the terms inside. This involves multiplying both $2x$ and $8$ by $-\frac{1}{4}$. This is a crucial step in simplifying the equation and bringing us closer to isolating the variable x. The distributive property states that a(b + c) = ab + ac. Applying this to our equation gives us:

$-\frac{1}{4} * (2x) + (-\frac{1}{4}) * 8 = -6$

Simplifying the multiplication, we get:

$-\frac{1}{2}x - 2 = -6$

Now, the equation looks simpler and easier to manipulate. We have successfully removed the parentheses and are ready to move on to the next step, which is isolating the term containing x.

Step 2: Isolate the Term with x

Our next objective is to isolate the term containing x, which is $-\frac{1}{2}x$. To do this, we need to eliminate the constant term, which is -2, from the left side of the equation. We can achieve this by adding 2 to both sides of the equation. This maintains the equality and brings us closer to isolating x. Remember, whatever operation we perform on one side of the equation, we must also perform on the other side to keep the equation balanced.

$-\frac{1}{2}x - 2 + 2 = -6 + 2$

This simplifies to:

$-\frac{1}{2}x = -4$

Now, we have successfully isolated the term with x on the left side of the equation. The next step will involve getting rid of the coefficient in front of x to finally solve for the variable.

Step 3: Solve for x

The final step in solving for x is to eliminate the coefficient $-\frac{1}{2}$ from the left side of the equation. To do this, we can multiply both sides of the equation by the reciprocal of $-\frac{1}{2}$, which is -2. This is the key to isolating x and finding its value. Multiplying both sides by -2 ensures that the equation remains balanced.

$-2 * (-\frac{1}{2}x) = -2 * (-4)$

This simplifies to:

$x = 8$

Therefore, the solution to the equation $-\frac{1}{4}(2x+8)=-6$ is x = 8. We have successfully solved the equation by following a series of algebraic steps.

Verification

To ensure the accuracy of our solution, it's always a good practice to verify the result by substituting the value of x back into the original equation. This confirmation step helps prevent errors and builds confidence in our solution. Let's substitute x = 8 into the original equation:

$-\frac{1}{4}(2 * 8 + 8) = -6$

Simplifying the expression inside the parentheses, we get:

$-\frac{1}{4}(16 + 8) = -6$

$-\frac{1}{4}(24) = -6$

$-6 = -6$

Since the left side of the equation equals the right side, our solution x = 8 is indeed correct. This verification step solidifies our understanding and confirms the accuracy of our calculations.

Conclusion

In this article, we have meticulously solved the linear equation $-\frac{1}{4}(2x+8)=-6$ by employing a step-by-step approach. We began by distributing the coefficient, then isolated the term with x, and finally solved for x. We also emphasized the importance of verifying the solution to ensure accuracy. This process highlights the fundamental principles of solving linear equations, which can be applied to a wide range of mathematical problems. Mastering these techniques is crucial for success in algebra and beyond. Understanding the underlying concepts and practicing regularly will make solving equations a natural and intuitive process.

By following these steps and understanding the underlying principles, you can confidently solve a variety of linear equations. Remember to practice regularly to solidify your skills and build your mathematical proficiency. Linear equations are a fundamental building block in mathematics, and mastering them will open doors to more advanced concepts and applications.