Find The Sum 6/x-4 5/x

Finding the Sum of 6/(x-4) and 5/x: A practical guide

This article provides a practical guide on how to find the sum of two algebraic fractions: 6/(x-4) and 5/x. We will explore the process step-by-step, covering the underlying mathematical principles and addressing common challenges encountered by students. Because of that, understanding this process is crucial for mastering algebraic manipulation and solving various mathematical problems involving fractions. Think about it: this guide aims to demystify the process, making it accessible to learners of all levels. We will look at the concepts of finding common denominators, simplifying expressions, and identifying restrictions on the variable x.

Understanding the Problem: Adding Fractions with Variables

Adding fractions, whether they contain numbers or variables, requires a common denominator. In this case, we are asked to find the sum of 6/(x-4) and 5/x. This is because we can only add or subtract fractions when they share the same denominator. The denominators are (x-4) and x, which are distinct. So, our first step is to find a common denominator that incorporates both (x-4) and x.

Step-by-Step Solution: Finding the Sum

1. Find the Least Common Denominator (LCD): The least common denominator is the smallest expression that is divisible by both denominators. In this instance, since (x-4) and x are distinct terms, the LCD is simply their product: x(x-4).

2. Rewrite the Fractions with the LCD: We need to rewrite both fractions so that they have the common denominator, x(x-4). To do this, we multiply each fraction by a cleverly chosen form of 1:

   • For the fraction 6/(x-4), we multiply by x/x:

     (6/(x-4)) * (x/x) = 6x / (x(x-4))

   • For the fraction 5/x, we multiply by (x-4)/(x-4):

     (5/x) * ((x-4)/(x-4)) = 5(x-4) / (x(x-4))

3. Add the Fractions: Now that both fractions have the same denominator, x(x-4), we can add their numerators:

   6x / (x(x-4)) + 5(x-4) / (x(x-4)) = (6x + 5(x-4)) / (x(x-4))

4. Simplify the Numerator: Expand and simplify the numerator:

   (6x + 5x - 20) / (x(x-4)) = (11x - 20) / (x(x-4))

5. Final Answer: The sum of 6/(x-4) and 5/x is (11x - 20) / (x(x-4)).

Explanation of the Mathematical Principles Involved

This problem exemplifies fundamental principles of algebraic manipulation:

• Finding the Least Common Multiple (LCM): The process of finding the least common denominator is closely related to finding the least common multiple (LCM) of two or more numbers or expressions. The LCM is the smallest number or expression that is a multiple of all the given numbers or expressions. In this case, the LCM of (x-4) and x is x(x-4).

• Equivalent Fractions: Multiplying the numerator and denominator of a fraction by the same non-zero value does not change the value of the fraction. This principle is crucial for rewriting fractions with a common denominator. We essentially multiplied each fraction by a form of 1 (x/x or (x-4)/(x-4)) to obtain equivalent fractions with the desired LCD.

• Adding Fractions: The rule for adding fractions with a common denominator is to add the numerators and keep the common denominator. This is a fundamental rule in arithmetic and algebra.

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• Simplifying Algebraic Expressions: After adding the fractions, the resulting expression often needs simplification. This typically involves expanding brackets, combining like terms, and factoring if possible.

Identifying Restrictions on the Variable x

It's crucial to identify any restrictions on the variable x. A restriction occurs when a value of x would make the denominator of the original fractions or the final result equal to zero. Division by zero is undefined in mathematics.

• Original Fractions: The original fractions are 6/(x-4) and 5/x. The denominator of the first fraction is (x-4), which is zero when x = 4. The denominator of the second fraction is x, which is zero when x = 0. Which means, x cannot be 0 or 4.

• Final Result: The denominator of the final result is x(x-4). This is zero when x = 0 or x = 4. This confirms our restrictions from the original fractions.

So, the final answer (11x - 20) / (x(x-4)) is valid for all real values of x except x = 0 and x = 4. We often express this as: x ∈ ℝ \ {0, 4} (meaning x belongs to the set of real numbers excluding 0 and 4).

Frequently Asked Questions (FAQ)

Q: Can I simplify the final expression further?

A: In this case, (11x - 20) / (x(x-4)) cannot be simplified further. There are no common factors between the numerator (11x - 20) and the denominator (x(x-4)).

Q: What if the denominators had common factors?

A: If the denominators had common factors, we would find the least common multiple (LCM) instead of simply multiplying them. Consider this: for example, if we were adding 2/(x-2) and 3/(2x-4), we would factor the second denominator as 2(x-2). The LCM would then be 2(x-2).

Q: What if the fractions had more terms in the denominator?

A: The process remains the same. You would still find the LCD by considering all factors in the denominators, even if they are more complex polynomials. The key is to find the smallest expression that contains all the factors present in all the denominators.

Q: What are some real-world applications of this type of problem?

A: Adding fractions with variables is frequently used in various fields, including:

• Physics: Solving problems involving resistances in parallel circuits.
• Engineering: Calculating the combined effect of multiple forces or components.
• Economics: Analyzing rates of change or combining different economic factors.
• Computer Science: Working with algorithms and data structures.

Conclusion

Adding fractions with variables, like 6/(x-4) and 5/x, involves finding a common denominator, rewriting the fractions, adding the numerators, and simplifying the resulting expression. Practically speaking, this process reinforces fundamental algebraic concepts, including finding the least common denominator, working with equivalent fractions, and simplifying expressions. On top of that, remember to always identify restrictions on the variable x to ensure the mathematical validity of your solution. This complete walkthrough provides a solid foundation for mastering algebraic manipulations involving fractions and addressing similar problems with confidence. Plus, by understanding the underlying principles and practicing regularly, you can develop proficiency in solving these kinds of algebraic problems. Remember to always double-check your work and ensure you've accounted for any restrictions on the variables.