Solve For X In The Equation 3(x-8) + 4x = 8x + 4

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In the realm of algebra, solving for unknown variables is a fundamental skill. This article delves into solving a specific linear equation: 3(x-8) + 4x = 8x + 4. We will systematically break down the steps involved, ensuring a clear understanding of the algebraic manipulations required to isolate the variable x and arrive at the correct solution. This process not only provides the answer but also reinforces the core principles of equation solving. Let's embark on this algebraic journey and master the art of finding the elusive x.

Understanding the Equation

Before diving into the solution, let's first understand the equation we are dealing with: 3(x-8) + 4x = 8x + 4. This is a linear equation in one variable, x. Our goal is to isolate x on one side of the equation to determine its value. This involves a series of algebraic operations, such as distribution, combining like terms, and using inverse operations to maintain the equality of the equation. The equation consists of two sides: the left-hand side (LHS) which is 3(x-8) + 4x, and the right-hand side (RHS) which is 8x + 4. The key to solving this equation is to manipulate both sides in a way that simplifies the expression while preserving the balance. We will meticulously dissect each step to ensure clarity and accuracy in our solution.

Step-by-Step Solution

To solve the equation 3(x-8) + 4x = 8x + 4, we will follow a step-by-step approach to ensure accuracy and clarity. This methodical process involves distributing, combining like terms, and isolating the variable x. Each step is crucial in simplifying the equation and moving closer to the solution. Let's begin with the first step:

Step 1: Distribute the 3

The first step in simplifying the equation is to distribute the 3 across the terms inside the parentheses on the left-hand side. This means multiplying 3 by both x and -8. This operation eliminates the parentheses and allows us to combine like terms in the subsequent steps. By applying the distributive property, we transform the equation into a more manageable form, setting the stage for further simplification.

3(x - 8) + 4x = 8x + 4 becomes 3x - 24 + 4x = 8x + 4

Step 2: Combine Like Terms

After distributing, the next step is to combine like terms on each side of the equation. Like terms are those that have the same variable raised to the same power. On the left-hand side, we have 3x and 4x, which can be combined. On the right-hand side, we only have 8x as a term with x and 4 as a constant term. Combining like terms simplifies the equation, making it easier to isolate the variable x. This step reduces the complexity of the equation, allowing us to proceed with the next phase of solving.

3x - 24 + 4x = 8x + 4 becomes 7x - 24 = 8x + 4

Step 3: Isolate the Variable Term

To isolate the variable term, we need to move all terms containing x to one side of the equation. A common approach is to subtract the smaller x term from both sides. In this case, we subtract 7x from both sides of the equation. This eliminates the x term from the left-hand side, moving us closer to isolating x. This strategic move simplifies the equation further, allowing us to focus on the constant terms in the next step.

7x - 24 = 8x + 4 becomes 7x - 24 - 7x = 8x + 4 - 7x which simplifies to -24 = x + 4

Step 4: Isolate the Constant Term

Now that we have the variable term on one side, we need to isolate the constant term on the other side. To do this, we subtract 4 from both sides of the equation. This eliminates the constant term on the right-hand side, leaving x isolated. This step completes the process of isolating the variable, leading us to the solution.

-24 = x + 4 becomes -24 - 4 = x + 4 - 4 which simplifies to -28 = x

The Solution

After completing the step-by-step solution, we arrive at the value of x. By carefully applying the principles of algebra, we have successfully isolated x and determined its value. The solution to the equation 3(x-8) + 4x = 8x + 4 is:

x = -28

This means that if we substitute -28 for x in the original equation, both sides of the equation will be equal. This confirms that our solution is correct and satisfies the equation. Solving for x is a fundamental skill in algebra, and this example demonstrates the systematic approach required to find the solution accurately.

Verification

To ensure the accuracy of our solution, we can verify it by substituting the value of x back into the original equation. This process confirms that our calculated value of x satisfies the equation and that our solution is correct. Substitution involves replacing x with -28 in the original equation and simplifying both sides to see if they are equal. This verification step is crucial in ensuring the reliability of our solution.

Substitute x = -28 into the original equation:

3(x - 8) + 4x = 8x + 4 becomes 3(-28 - 8) + 4(-28) = 8(-28) + 4

Simplify both sides:

Left-hand side (LHS): 3(-36) - 112 = -108 - 112 = -220 Right-hand side (RHS): -224 + 4 = -220

Since the LHS equals the RHS, our solution is verified:

-220 = -220

This confirms that x = -28 is indeed the correct solution to the equation.

Common Mistakes to Avoid

When solving equations like 3(x-8) + 4x = 8x + 4, it's essential to be aware of common mistakes that can lead to incorrect solutions. Avoiding these pitfalls will help ensure accuracy and proficiency in solving algebraic equations. Here are some frequent errors to watch out for:

  1. Incorrect Distribution: A common mistake is failing to distribute the number outside the parentheses correctly. For instance, in the first step, it's crucial to multiply 3 by both x and -8. Incorrectly distributing can lead to a flawed equation and an incorrect solution.
  2. Combining Unlike Terms: Another frequent error is combining terms that are not like terms. Only terms with the same variable and exponent can be combined. For example, 3x and 4x can be combined, but 3x and 4 cannot.
  3. Sign Errors: Sign errors are common, especially when dealing with negative numbers. Ensure that you correctly apply the rules of addition, subtraction, multiplication, and division with negative numbers. A single sign error can throw off the entire solution.
  4. Incorrect Order of Operations: Following the correct order of operations (PEMDAS/BODMAS) is crucial. Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Failing to follow this order can lead to an incorrect simplification and solution.
  5. Not Performing the Same Operation on Both Sides: To maintain the equality of the equation, any operation performed on one side must also be performed on the other side. Forgetting to do this can disrupt the balance of the equation and lead to an incorrect answer.

By being mindful of these common mistakes, you can enhance your accuracy and confidence in solving algebraic equations.

Alternative Approaches

While the step-by-step solution outlined above is a standard approach, there can be alternative methods to solve the equation 3(x-8) + 4x = 8x + 4. Exploring these different approaches can enhance your problem-solving skills and provide a deeper understanding of algebraic manipulations. Here are a couple of alternative strategies:

  1. Rearranging Terms Differently: Instead of immediately distributing, you could rearrange the terms on the left-hand side first. For example, you could rewrite the equation as 4x + 3(x - 8) = 8x + 4. While this doesn't change the fundamental steps, it might provide a different perspective on the equation.
  2. Isolating Terms Differently: In the step where we isolated the variable term, we subtracted 7x from both sides. Alternatively, you could subtract 8x from both sides. This would result in a negative coefficient for x, but it's a valid approach that will still lead to the correct solution if handled carefully.

No matter the approach, the key is to follow the rules of algebra and maintain the equality of the equation. By exploring different methods, you can develop a more flexible and comprehensive understanding of equation solving.

Conclusion

In conclusion, solving for x in the equation 3(x-8) + 4x = 8x + 4 involves a systematic approach that includes distribution, combining like terms, and isolating the variable. By carefully following these steps, we arrived at the solution x = -28. We also verified this solution by substituting it back into the original equation and confirming that both sides are equal. Furthermore, we discussed common mistakes to avoid and explored alternative approaches to solving the equation. Mastering these techniques enhances your algebraic skills and provides a solid foundation for tackling more complex equations. Remember, practice and attention to detail are key to success in algebra.

Therefore, the correct answer is 4) -28