Simplify 3 5 X − 2 X − 1 X \frac{3}{5x} - \frac{2x-1}{x} 5 X 3 ​ − X 2 X − 1 ​ .

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is often required to solve various mathematical problems. In this article, we will focus on simplifying the given expression $\frac{3}{5x} - \frac{2x-1}{x}$. This expression involves fractions and variables, and we will use various techniques to simplify it.

Understanding the Expression

The given expression is a combination of two fractions: $\frac{3}{5x}$ and $\frac{2x-1}{x}$. To simplify this expression, we need to find a common denominator for both fractions. The common denominator is the least common multiple (LCM) of the denominators of both fractions.

Finding the Common Denominator

The denominator of the first fraction is $5x$, and the denominator of the second fraction is $x$. To find the LCM of $5x$ and $x$, we need to consider the prime factors of both numbers. The prime factorization of $5x$ is $5 \cdot x$, and the prime factorization of $x$ is $x$. Since $x$ is a common factor, we can cancel it out, leaving us with $5$ as the LCM.

Simplifying the Expression

Now that we have found the common denominator, we can rewrite both fractions with the common denominator. The first fraction becomes $\frac{3}{5x} = \frac{3x}{5x^2}$, and the second fraction becomes $\frac{2x-1}{x} = \frac{2x^2 - x}{5x^2}$. Now we can subtract the two fractions.

Subtracting the Fractions

To subtract the fractions, we need to have the same denominator for both fractions. We already have the common denominator, which is $5x^2$. Now we can subtract the numerators: $\frac{3x}{5x^2} - \frac{2x^2 - x}{5x^2} = \frac{3x - (2x^2 - x)}{5x^2}$.

Simplifying the Numerator

Now we can simplify the numerator by combining like terms: $3x - 2x^2 + x = -2x^2 + 4x$. So the expression becomes $\frac{-2x^2 + 4x}{5x^2}$.

Canceling Out Common Factors

We can simplify the expression further by canceling out common factors. The numerator and denominator both have a common factor of $x$. We can cancel out this factor, leaving us with $\frac{-2x + 4}{5x}$.

Final Simplification

The expression $\frac{-2x + 4}{5x}$ can be simplified further by factoring out a common factor of $2$ from the numerator: $\frac{-2(x - 2)}{5x}$. This is the final simplified form of the expression.

Conclusion

In this article, we simplified the expression $\frac{3}{5x} - \frac{2x-1}{x}$ using various techniques. We found the common denominator, rewrote both fractions with the common denominator, subtracted the fractions, simplified the numerator, canceled out common factors, and finally simplified the expression to its final form. This expression is now in a simpler form, making it easier to work with in mathematical problems.

Example Use Case

The simplified expression $\frac{-2(x - 2)}{5x}$ can be used in various mathematical problems, such as solving equations or graphing functions. For example, if we have an equation $y = \frac{-2(x - 2)}{5x}$, we can use this expression to find the value of $y$ for a given value of $x$.

Tips and Tricks

When simplifying algebraic expressions, it is essential to find the common denominator and rewrite both fractions with the common denominator. This will make it easier to subtract the fractions and simplify the expression. Additionally, canceling out common factors can help simplify the expression further.

Common Mistakes

When simplifying algebraic expressions, it is common to make mistakes such as forgetting to find the common denominator or canceling out common factors incorrectly. To avoid these mistakes, it is essential to carefully read and follow the steps outlined in this article.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it is often required to solve various mathematical problems. In this article, we simplified the expression $\frac{3}{5x} - \frac{2x-1}{x}$ using various techniques. By following the steps outlined in this article, you can simplify algebraic expressions and make them easier to work with in mathematical problems.

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Introduction

In our previous article, we simplified the expression $\frac{3}{5x} - \frac{2x-1}{x}$ using various techniques. In this article, we will answer some common questions related to simplifying algebraic expressions, including the one we simplified in our previous article.

Q: What is the common denominator of two fractions?

A: The common denominator of two fractions is the least common multiple (LCM) of the denominators of both fractions. To find the LCM, we need to consider the prime factors of both numbers.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, we need to consider the prime factors of both numbers. We can use the following steps:

1. Find the prime factors of both numbers.
2. Identify the common prime factors.
3. Multiply the common prime factors by the remaining prime factors to find the LCM.

Q: What is the difference between a common denominator and a common factor?

A: A common denominator is the least common multiple (LCM) of the denominators of two fractions, while a common factor is a factor that is common to both numbers.

Q: How do I simplify a fraction with a variable in the denominator?

A: To simplify a fraction with a variable in the denominator, we need to find the common denominator and rewrite both fractions with the common denominator. We can then simplify the expression by canceling out common factors.

Q: What is the final simplified form of the expression $\frac{3}{5x} - \frac{2x-1}{x}$?

A: The final simplified form of the expression $\frac{3}{5x} - \frac{2x-1}{x}$ is $\frac{-2(x - 2)}{5x}$.

Q: How do I use the simplified expression in a mathematical problem?

A: We can use the simplified expression $\frac{-2(x - 2)}{5x}$ in various mathematical problems, such as solving equations or graphing functions. For example, if we have an equation $y = \frac{-2(x - 2)}{5x}$, we can use this expression to find the value of $y$ for a given value of $x$.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

• Forgetting to find the common denominator
• Canceling out common factors incorrectly
• Not simplifying the expression fully

Q: How can I practice simplifying algebraic expressions?

A: We can practice simplifying algebraic expressions by working on various problems and exercises. We can also use online resources and tools to help us practice and improve our skills.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

• Solving equations and inequalities
• Graphing functions and curves
• Modeling real-world phenomena
• Optimizing systems and processes

Conclusion

In this article, we answered some common questions related to simplifying algebraic expressions, including the one we simplified in our previous article. We hope that this article has been helpful in providing you with a better understanding of simplifying algebraic expressions and how to apply this skill in various mathematical problems.

Example Use Case

The simplified expression $\frac{-2(x - 2)}{5x}$ can be used in various mathematical problems, such as solving equations or graphing functions. For example, if we have an equation $y = \frac{-2(x - 2)}{5x}$, we can use this expression to find the value of $y$ for a given value of $x$.

Tips and Tricks

When simplifying algebraic expressions, it is essential to find the common denominator and rewrite both fractions with the common denominator. This will make it easier to subtract the fractions and simplify the expression. Additionally, canceling out common factors can help simplify the expression further.

Common Mistakes

When simplifying algebraic expressions, it is common to make mistakes such as forgetting to find the common denominator or canceling out common factors incorrectly. To avoid these mistakes, it is essential to carefully read and follow the steps outlined in this article.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it is often required to solve various mathematical problems. By following the steps outlined in this article, you can simplify algebraic expressions and make them easier to work with in mathematical problems.