Write The Expression $\frac{3a}{2a-1} - \frac{1}{3a+1}$ As One Fraction.

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Introduction

In this article, we will explore the process of combining two fractions into a single fraction. This is a fundamental concept in algebra and is essential for solving various mathematical problems. We will use the given expression 3a2a−1−13a+1\frac{3a}{2a-1} - \frac{1}{3a+1} as an example and demonstrate how to simplify it into a single fraction.

Understanding the Problem

The given expression is a difference of two fractions, and our goal is to combine them into a single fraction. To do this, 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 denominators of the two fractions are 2a−12a-1 and 3a+13a+1. To find the LCM of these two expressions, we can list the multiples of each expression and find the smallest multiple that is common to both.

  • The multiples of 2a−12a-1 are: (2a−1),2(2a−1),3(2a−1),…(2a-1), 2(2a-1), 3(2a-1), \ldots
  • The multiples of 3a+13a+1 are: (3a+1),2(3a+1),3(3a+1),…(3a+1), 2(3a+1), 3(3a+1), \ldots

By examining the lists, we can see that the smallest multiple that is common to both is (2a−1)(3a+1)(2a-1)(3a+1).

Simplifying the Expression

Now that we have found the common denominator, we can rewrite each fraction with the common denominator.

3a2a−1−13a+1=3a(3a+1)(2a−1)(3a+1)−(2a−1)(2a−1)(3a+1)\frac{3a}{2a-1} - \frac{1}{3a+1} = \frac{3a(3a+1)}{(2a-1)(3a+1)} - \frac{(2a-1)}{(2a-1)(3a+1)}

Next, we can simplify each fraction by canceling out any common factors in the numerator and denominator.

3a(3a+1)(2a−1)(3a+1)−(2a−1)(2a−1)(3a+1)=3a(3a+1)−(2a−1)(2a−1)(3a+1)\frac{3a(3a+1)}{(2a-1)(3a+1)} - \frac{(2a-1)}{(2a-1)(3a+1)} = \frac{3a(3a+1) - (2a-1)}{(2a-1)(3a+1)}

Expanding and Simplifying

To simplify the expression further, we can expand the numerator and combine like terms.

3a(3a+1)−(2a−1)(2a−1)(3a+1)=9a2+3a−2a+1(2a−1)(3a+1)\frac{3a(3a+1) - (2a-1)}{(2a-1)(3a+1)} = \frac{9a^2 + 3a - 2a + 1}{(2a-1)(3a+1)}

9a2+3a−2a+1(2a−1)(3a+1)=9a2+a+1(2a−1)(3a+1)\frac{9a^2 + 3a - 2a + 1}{(2a-1)(3a+1)} = \frac{9a^2 + a + 1}{(2a-1)(3a+1)}

Final Answer

The final answer is 9a2+a+1(2a−1)(3a+1)\boxed{\frac{9a^2 + a + 1}{(2a-1)(3a+1)}}.

Conclusion

In this article, we have demonstrated how to combine two fractions into a single fraction. We used the given expression 3a2a−1−13a+1\frac{3a}{2a-1} - \frac{1}{3a+1} as an example and simplified it into a single fraction. The process involved finding the common denominator, rewriting each fraction with the common denominator, and simplifying the expression by canceling out common factors and combining like terms. This is a fundamental concept in algebra and is essential for solving various mathematical problems.

Common Applications

The concept of combining fractions is used in various mathematical applications, including:

  • Algebra: Combining fractions is a fundamental concept in algebra and is used to simplify expressions and solve equations.
  • Calculus: Combining fractions is used in calculus to simplify expressions and solve problems involving limits, derivatives, and integrals.
  • Physics: Combining fractions is used in physics to simplify expressions and solve problems involving motion, energy, and momentum.

Tips and Tricks

Here are some tips and tricks for combining fractions:

  • Find the common denominator: The common denominator is the least common multiple (LCM) of the denominators of both fractions.
  • Rewrite each fraction with the common denominator: Once you have found the common denominator, rewrite each fraction with the common denominator.
  • Simplify the expression: Simplify the expression by canceling out common factors and combining like terms.

By following these tips and tricks, you can simplify expressions and solve problems involving fractions with ease.

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Q: What is the first step in combining fractions?

A: The first step in combining fractions is to find the common denominator. The common denominator is the least common multiple (LCM) of the denominators of both fractions.

Q: How do I find the common denominator?

A: To find the common denominator, you can list the multiples of each expression and find the smallest multiple that is common to both. Alternatively, you can use the formula: LCM(a, b) = (a × b) / GCD(a, b), where GCD(a, b) is the greatest common divisor of a and b.

Q: What is the next step after finding the common denominator?

A: After finding the common denominator, you need to rewrite each fraction with the common denominator. This involves multiplying the numerator and denominator of each fraction by the necessary factors to obtain the common denominator.

Q: How do I simplify the expression after combining fractions?

A: To simplify the expression after combining fractions, you need to cancel out any common factors in the numerator and denominator. This involves dividing both the numerator and denominator by the greatest common factor.

Q: What are some common mistakes to avoid when combining fractions?

A: Some common mistakes to avoid when combining fractions include:

  • Not finding the common denominator: Failing to find the common denominator can lead to incorrect results.
  • Not rewriting each fraction with the common denominator: Failing to rewrite each fraction with the common denominator can lead to incorrect results.
  • Not simplifying the expression: Failing to simplify the expression can lead to unnecessary complexity.

Q: How do I check my work when combining fractions?

A: To check your work when combining fractions, you can:

  • Verify the common denominator: Check that the common denominator is correct.
  • Verify the rewritten fractions: Check that each fraction has been rewritten with the correct common denominator.
  • Verify the simplified expression: Check that the expression has been simplified correctly.

Q: What are some real-world applications of combining fractions?

A: Combining fractions has many real-world applications, including:

  • Cooking: Combining fractions is used in cooking to measure ingredients and proportions.
  • Finance: Combining fractions is used in finance to calculate interest rates and investment returns.
  • Science: Combining fractions is used in science to calculate concentrations and proportions.

Q: How do I practice combining fractions?

A: To practice combining fractions, you can:

  • Work through examples: Practice combining fractions by working through examples and exercises.
  • Use online resources: Use online resources, such as calculators and worksheets, to practice combining fractions.
  • Seek help: Seek help from a teacher or tutor if you are struggling with combining fractions.

Q: What are some common misconceptions about combining fractions?

A: Some common misconceptions about combining fractions include:

  • Believing that combining fractions is only for simple fractions: Combining fractions can be used for complex fractions as well.
  • Believing that combining fractions is only for addition: Combining fractions can be used for subtraction and other operations as well.
  • Believing that combining fractions is only for basic arithmetic: Combining fractions can be used for advanced arithmetic and algebra as well.

By understanding these common misconceptions, you can avoid making mistakes and improve your skills in combining fractions.