Solving X + 5x + 5(5x) + 2(5(5x)) = 81 A Comprehensive Guide

This article provides a comprehensive guide on how to solve the equation x + 5x + 5(5x) + 2(5(5x)) = 81. We will break down the steps involved in simplifying the equation, combining like terms, and isolating the variable x to find the solution. This detailed explanation is designed to help students and anyone interested in algebra understand the process thoroughly. By the end of this guide, you will have a clear understanding of how to solve similar equations and improve your algebra skills.

Understanding the Equation

Before diving into the solution, it's crucial to understand the equation: x + 5x + 5(5x) + 2(5(5x)) = 81. This equation involves a variable x and several terms that need to be simplified and combined. The goal is to isolate x on one side of the equation to determine its value. This involves several algebraic steps, including distribution, combining like terms, and performing inverse operations. A clear understanding of these steps is essential for solving not only this equation but also other algebraic problems.

Breaking Down the Terms

Let’s break down each term in the equation to better understand how to simplify it:

• x: This is the variable we need to solve for.
• 5x: This term is a multiple of x, which can be combined with other x terms.
• 5(5x): This term requires us to multiply 5 by 5x, which simplifies to 25x.
• 2(5(5x)): This term requires us to multiply 2 by 5(5x), which simplifies to 2 multiplied by 25x, resulting in 50x.

By understanding each term, we can proceed with simplifying the equation by combining like terms. This initial breakdown is a critical step in solving any algebraic equation.

Step-by-Step Solution

Now, let’s walk through the step-by-step solution to the equation x + 5x + 5(5x) + 2(5(5x)) = 81. Each step is explained in detail to ensure clarity and understanding.

Step 1: Simplify the Terms

First, we simplify the terms involving multiplication:

• 5(5x) = 25x
• 2(5(5x)) = 2(25x) = 50x

So, the equation becomes: x + 5x + 25x + 50x = 81. This simplification makes the equation easier to work with by reducing the number of operations needed.

Step 2: Combine Like Terms

Next, we combine the like terms on the left side of the equation. Like terms are terms that contain the same variable raised to the same power. In this case, all terms contain x to the power of 1, so they can be combined:

• x + 5x + 25x + 50x = (1 + 5 + 25 + 50)x

Adding the coefficients (the numbers in front of x), we get:

• (1 + 5 + 25 + 50)x = 81x

So, the equation is now: 81x = 81. This step significantly simplifies the equation, making it easier to solve for x.

Step 3: Isolate the Variable

To isolate the variable x, we need to undo the multiplication by 81. We do this by dividing both sides of the equation by 81:

• 81x / 81 = 81 / 81

This simplifies to:

• x = 1

Therefore, the solution to the equation is x = 1. This final step is crucial for finding the value of x and completing the solution process.

Detailed Explanation of Each Step

To further clarify the solution process, let's delve into a detailed explanation of each step. Understanding the reasoning behind each step is crucial for mastering algebraic problem-solving.

Detailed Explanation of Step 1: Simplify the Terms

Simplifying terms in an equation is a fundamental step in algebra. It involves performing operations such as multiplication and division to reduce the complexity of the equation. In our case, we had terms like 5(5x) and 2(5(5x)). To simplify these, we need to follow the order of operations (PEMDAS/BODMAS), which dictates that we perform multiplication before addition or subtraction.

• For 5(5x), we multiply 5 by 5x to get 25x. This is a straightforward application of the associative property of multiplication.
• For 2(5(5x)), we first simplify the inner parentheses, 5(5x), which we already know is 25x. Then, we multiply 2 by 25x to get 50x. This step-by-step simplification makes the equation easier to manage.

By simplifying these terms, we reduce the number of operations in the equation, making it easier to combine like terms in the next step. This process is essential for efficiently solving algebraic equations.

Detailed Explanation of Step 2: Combine Like Terms

Combining like terms is another critical step in simplifying equations. Like terms are terms that have the same variable raised to the same power. In our equation, x, 5x, 25x, and 50x are all like terms because they all contain the variable x raised to the power of 1. To combine them, we add their coefficients (the numbers in front of the variable).

• The coefficient of x is 1 (since x is the same as 1x).
• The coefficient of 5x is 5.
• The coefficient of 25x is 25.
• The coefficient of 50x is 50.

Adding these coefficients together, we get 1 + 5 + 25 + 50 = 81. Therefore, the combined term is 81x. This step condenses the equation, making it much simpler to solve. Combining like terms is a fundamental skill in algebra and is used extensively in solving various types of equations.

Detailed Explanation of Step 3: Isolate the Variable

Isolating the variable is the final step in solving for x. This means getting x by itself on one side of the equation. In our equation, we have 81x = 81. To isolate x, we need to undo the multiplication by 81. The inverse operation of multiplication is division, so we divide both sides of the equation by 81.

• Dividing both sides by 81, we get (81x) / 81 = 81 / 81.
• On the left side, 81x divided by 81 simplifies to x.
• On the right side, 81 divided by 81 is 1.

Therefore, we have x = 1. This step demonstrates the principle of maintaining equality by performing the same operation on both sides of the equation. Isolating the variable is the key to finding the solution and is a core concept in algebra.

Common Mistakes to Avoid

When solving equations, it’s easy to make mistakes if you’re not careful. Here are some common mistakes to avoid:

1. Incorrectly Distributing: Make sure to distribute multiplication over all terms inside parentheses. For example, in 2(5(5x)), ensure you multiply correctly.
2. Combining Unlike Terms: Only combine terms that have the same variable and exponent. For instance, you can combine 5x and 25x, but not 5x and 25x². Understanding this difference is crucial for accurate simplification.
3. Forgetting to Perform Operations on Both Sides: When isolating the variable, remember to perform the same operation on both sides of the equation to maintain balance. If you divide one side by a number, you must divide the other side by the same number. Neglecting this can lead to incorrect solutions.
4. Arithmetic Errors: Simple addition, subtraction, multiplication, or division errors can lead to incorrect answers. Double-check your calculations to ensure accuracy. Using a calculator can help reduce these errors.

By being aware of these common mistakes and taking steps to avoid them, you can increase your accuracy and confidence in solving algebraic equations.

Practice Problems

To solidify your understanding, try solving these practice problems. Each problem is similar to the one we solved in this guide.

1. Solve: 2x + 4x + 3(4x) + 2(3(4x)) = 100
2. Solve: y + 3y + 2(3y) + 4(2(3y)) = 78
3. Solve: 3z + 6z + 4(6z) + 2(4(6z)) = 165

Working through these problems will give you valuable practice and reinforce the steps involved in solving algebraic equations. Take your time, follow the steps outlined in this guide, and check your answers carefully.

Conclusion

In conclusion, solving the equation x + 5x + 5(5x) + 2(5(5x)) = 81 involves several key steps: simplifying terms, combining like terms, and isolating the variable. By understanding each step and the principles behind them, you can confidently solve similar algebraic equations. Remember to avoid common mistakes and practice regularly to improve your skills. Algebra is a fundamental branch of mathematics, and mastering it will open doors to more advanced topics and real-world applications. Keep practicing, and you’ll become proficient in solving equations and other algebraic problems.