Solve The Equation: 25 X − 2 − 8 X = 15 X \frac{25}{x-2} - \frac{8}{x} = \frac{15}{x} X − 2 25 − X 8 = X 15

Mar 13, 2025 by ADMIN 114 views

**Solving the Equation: A Step-by-Step Guide**

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Introduction

In this article, we will delve into the world of algebra and focus on solving a complex equation involving fractions. The equation we will be tackling is $\frac{25}{x-2} - \frac{8}{x} = \frac{15}{x}$ . This equation may seem daunting at first, but with a systematic approach and a clear understanding of algebraic concepts, we can break it down and find the solution.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its components. We have three fractions, each with a different denominator. The first fraction has a denominator of $x-2$ , the second fraction has a denominator of $x$ , and the third fraction also has a denominator of $x$ . Our goal is to find the value of $x$ that satisfies the equation.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To simplify the equation and eliminate the fractions, we need to multiply both sides by the least common multiple (LCM) of the denominators. In this case, the LCM is $x(x-2)$ . By multiplying both sides by $x(x-2)$ , we can eliminate the fractions and work with a simpler equation.

\frac{25}{x-2} \cdot x(x-2) - \frac{8}{x} \cdot x(x-2) = \frac{15}{x} \cdot x(x-2)

Simplifying the Equation

After multiplying both sides by the LCM, we can simplify the equation by expanding the products.

25x - 50 - 8(x-2) = 15(x-2)

Distributing and Combining Like Terms

Next, we need to distribute the terms on the left-hand side and combine like terms.

25x - 50 - 8x + 16 = 15x - 30

Simplifying Further

We can simplify the equation further by combining like terms.

17x - 34 = 15x - 30

Isolating the Variable

Our goal is to isolate the variable $x$ on one side of the equation. To do this, we can add $34$ to both sides of the equation.

17x = 15x + 4

Subtracting 15x from Both Sides

Next, we can subtract $15x$ from both sides of the equation to isolate the term with $x$ .

2x = 4

Dividing Both Sides by 2

Finally, we can divide both sides of the equation by $2$ to find the value of $x$ .

x = 2

Conclusion

In this article, we solved the equation $\frac{25}{x-2} - \frac{8}{x} = \frac{15}{x}$ using a step-by-step approach. We multiplied both sides by the least common multiple, simplified the equation, distributed and combined like terms, and finally isolated the variable $x$ . The solution to the equation is $x = 2$ .

Real-World Applications

Solving equations like this one has many real-world applications. In physics, for example, equations like this one can be used to model the motion of objects. In engineering, equations like this one can be used to design and optimize systems. In finance, equations like this one can be used to model and analyze financial markets.

Tips and Tricks

When solving equations like this one, it's essential to follow a systematic approach. Start by identifying the least common multiple of the denominators, then multiply both sides by the LCM. Simplify the equation, distribute and combine like terms, and finally isolate the variable. With practice and patience, you can become proficient in solving equations like this one.

Common Mistakes

When solving equations like this one, it's easy to make mistakes. Some common mistakes include:

Not identifying the least common multiple: Make sure to identify the LCM of the denominators before multiplying both sides by it.
Not simplifying the equation: Make sure to simplify the equation after multiplying both sides by the LCM.
Not distributing and combining like terms: Make sure to distribute and combine like terms after simplifying the equation.
Not isolating the variable: Make sure to isolate the variable $x$ on one side of the equation.

Practice Problems

To practice solving equations like this one, try the following problems:

Problem 1: Solve the equation $\frac{10}{x+1} - \frac{5}{x} = \frac{3}{x}$ .
Problem 2: Solve the equation $\frac{20}{x-1} - \frac{10}{x} = \frac{5}{x}$ .
Problem 3: Solve the equation $\frac{15}{x+2} - \frac{8}{x} = \frac{4}{x}$ .

Conclusion

Solving equations like $\frac{25}{x-2} - \frac{8}{x} = \frac{15}{x}$ requires a systematic approach and a clear understanding of algebraic concepts. By following the steps outlined in this article, you can become proficient in solving equations like this one. Remember to identify the least common multiple, simplify the equation, distribute and combine like terms, and finally isolate the variable. With practice and patience, you can become a master of solving equations like this one.

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Introduction

In our previous article, we solved the equation $\frac{25}{x-2} - \frac{8}{x} = \frac{15}{x}$ using a step-by-step approach. In this article, we will answer some of the most frequently asked questions about solving equations like this one.

Q: What is the least common multiple (LCM) of the denominators?

A: The least common multiple (LCM) of the denominators is the smallest multiple that all the denominators have in common. In the case of the equation $\frac{25}{x-2} - \frac{8}{x} = \frac{15}{x}$ , the LCM is $x(x-2)$ .

Q: Why do we need to multiply both sides by the LCM?

A: We need to multiply both sides by the LCM to eliminate the fractions and simplify the equation. By multiplying both sides by the LCM, we can work with a simpler equation and make it easier to solve.

Q: How do I identify the LCM of the denominators?

A: To identify the LCM of the denominators, you need to list the multiples of each denominator and find the smallest multiple that they all have in common. For example, if the denominators are $x-2$ and $x$ , the multiples of $x-2$ are $x-2$ , $2x-4$ , $3x-6$ , and so on. The multiples of $x$ are $x$ , $2x$ , $3x$ , and so on. The smallest multiple that they all have in common is $x(x-2)$ .

Q: What if the LCM is not a simple expression?

A: If the LCM is not a simple expression, you may need to use a different method to solve the equation. For example, you can try to simplify the equation by combining like terms or using a different algebraic technique.

Q: Can I use a calculator to solve equations like this one?

A: Yes, you can use a calculator to solve equations like this one. However, it's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

Q: What if I get stuck while solving an equation?

A: If you get stuck while solving an equation, don't be afraid to ask for help. You can ask a teacher, a tutor, or a classmate for assistance. You can also try to break the equation down into smaller parts and solve each part separately.

Q: How do I know if I have solved the equation correctly?

A: To know if you have solved the equation correctly, you need to check your work by plugging the solution back into the original equation. If the solution satisfies the equation, then you have solved it correctly.

Q: Can I use equations like this one to model real-world problems?

A: Yes, you can use equations like this one to model real-world problems. For example, you can use the equation $\frac{25}{x-2} - \frac{8}{x} = \frac{15}{x}$ to model the motion of an object or the flow of a fluid.

Q: What are some common mistakes to avoid when solving equations like this one?

A: Some common mistakes to avoid when solving equations like this one include:

Not identifying the LCM: Make sure to identify the LCM of the denominators before multiplying both sides by it.
Not simplifying the equation: Make sure to simplify the equation after multiplying both sides by the LCM.
Not distributing and combining like terms: Make sure to distribute and combine like terms after simplifying the equation.
Not isolating the variable: Make sure to isolate the variable $x$ on one side of the equation.