Solving $\frac{15}{x-6}+\frac{7}{x+6}$ Equivalent Expressions Guide
In the realm of algebra, equivalent expressions are mathematical phrases that, while appearing different, hold the same value for every possible value of the variable. This article delves into the process of identifying equivalent expressions, focusing on the specific problem of simplifying the sum of two fractions: $ rac{15}{x-6}+ rac{7}{x+6}$. Understanding how to manipulate and simplify such expressions is a fundamental skill in algebra, crucial for solving equations, simplifying complex formulas, and tackling more advanced mathematical concepts.
Problem Statement Unpacking the Expression
The question at hand asks us to find an expression equivalent to $ rac{15}{x-6}+ rac{7}{x+6}$, given that no denominator equals zero. This condition is crucial because division by zero is undefined in mathematics. The problem essentially requires us to add two fractions with different denominators, a common task in algebraic simplification. To accomplish this, we need to find a common denominator, combine the fractions, and then simplify the resulting expression.
The ability to work with algebraic fractions is essential in various mathematical contexts. From solving rational equations to simplifying complex algebraic expressions, the techniques used here form the bedrock of many mathematical problem-solving strategies. This article will provide a step-by-step guide on how to approach such problems, ensuring a clear understanding of the underlying principles.
Step-by-Step Solution Finding the Common Denominator
The first step in adding fractions is to find a common denominator. In this case, we have two fractions with denominators $x-6$ and $x+6$. The least common denominator (LCD) is the product of these two denominators, which is $(x-6)(x+6)$. This is because $x-6$ and $x+6$ do not share any common factors, making their product the smallest expression that both denominators can divide into evenly.
Finding the least common denominator is a critical step in adding or subtracting fractions. It ensures that we are working with fractions that have the same "size" of pieces, allowing us to combine them accurately. In this specific problem, recognizing that the denominators are binomials (expressions with two terms) and that they are conjugates (differing only in the sign between the terms) simplifies the process of finding the LCD.
Combining the Fractions Multiplying Numerators and Denominators
Once we have the common denominator, we need to rewrite each fraction with this denominator. To do this, we multiply the numerator and denominator of each fraction by the factor that will result in the common denominator. For the first fraction, $ rac{15}{x-6}$, we multiply both the numerator and the denominator by $x+6$:
Similarly, for the second fraction, $\frac{7}{x+6}$, we multiply both the numerator and the denominator by $x-6$:
Now that both fractions have the same denominator, we can add them together. This involves adding the numerators and keeping the common denominator:
This step is crucial in combining fractions. By ensuring that both fractions have the same denominator, we can simply add the numerators, making the addition process straightforward. The multiplication of the numerators and denominators is a direct application of the principle that multiplying a fraction by a form of 1 (in this case, $\frac{x+6}{x+6}$ and $\frac{x-6}{x-6}$) does not change its value.
Simplifying the Expression Expanding and Combining Like Terms
Next, we simplify the numerator by expanding the products and combining like terms:
Combining like terms (the terms with $x$ and the constant terms), we get:
So, the numerator simplifies to $22x + 48$.
Now, let's look at the denominator. We have $(x-6)(x+6)$. This is a special product known as the difference of squares, which can be expanded as:
Therefore, the expression becomes:
Simplifying algebraic expressions is a key skill in algebra. It involves expanding products, combining like terms, and recognizing special products like the difference of squares. The goal is to write the expression in its simplest form, making it easier to understand and work with. In this case, expanding the products in the numerator and denominator and then combining like terms led us to the simplified expression.
Identifying the Correct Answer Matching the Simplified Expression
Now we compare our simplified expression, $\frac{22x + 48}{x^2 - 36}$, with the given options:
A. $\frac{22 x+132}{x^2-36}$ B. $\frac{22 x+48}{x^2-36}$ C. $\frac{22}{x^2-36}$ D. $ rac{22 x-48}{x^2-36}$
Option B, $\frac{22 x+48}{x^2-36}$, matches our simplified expression.
This step of identifying the correct answer is crucial. It involves comparing the simplified expression with the given options and selecting the one that matches. This requires careful attention to detail and a thorough understanding of the simplification process.
Common Mistakes to Avoid Errors in Simplification
When working with algebraic fractions, there are several common mistakes to watch out for. One frequent error is incorrectly finding the common denominator. It's essential to ensure that the LCD is the least common multiple of the denominators.
Another common mistake is errors in expanding and simplifying expressions. For example, students might incorrectly distribute a number over parentheses or fail to combine like terms correctly. It's important to take each step carefully and double-check your work.
Additionally, mistakes can occur when dealing with negative signs. Ensure that you distribute negative signs correctly when expanding expressions.
Being aware of these common mistakes can help prevent errors and improve accuracy in solving algebraic problems. It's always a good practice to double-check each step and to be mindful of potential pitfalls.
Alternative Approaches Exploring Different Methods
While the step-by-step solution outlined above is a standard approach, there might be alternative methods to solve this problem. One approach could involve working backwards from the answer choices. By manipulating the answer choices, you might be able to see which one simplifies to the original expression.
Another approach could involve using technology, such as a computer algebra system (CAS), to simplify the expression. CAS tools can quickly perform algebraic manipulations and simplify complex expressions.
Exploring alternative approaches can provide a deeper understanding of the problem and can sometimes lead to more efficient solutions. It also reinforces the idea that there are often multiple ways to solve a mathematical problem.
Conclusion Key Takeaways and Problem-Solving Strategies
In summary, solving the problem $\frac{15}{x-6}+\frac{7}{x+6}$ involves finding a common denominator, combining the fractions, and simplifying the resulting expression. The correct answer is $\frac{22x + 48}{x^2 - 36}$, which corresponds to option B.
Key takeaways from this problem include:
- The importance of finding the least common denominator when adding fractions.
- The process of multiplying the numerator and denominator by the appropriate factors to obtain the common denominator.
- The need to expand and simplify expressions carefully, paying attention to signs and like terms.
- The recognition of special products like the difference of squares.
By mastering these concepts and techniques, you can confidently tackle similar algebraic problems. Remember to practice regularly and to double-check your work to avoid common mistakes.
Problem-solving strategies like these are essential for success in algebra and beyond. They provide a structured approach to tackling complex problems and help build a strong foundation in mathematical thinking.
Final Answer Option B is Correct
Therefore, the expression equivalent to $\frac{15}{x-6}+\frac{7}{x+6}$ is $\frac{22 x+48}{x^2-36}$. Option B is the correct answer.
Equivalent expressions, Algebraic fractions, Least common denominator, Simplifying algebraic expressions, Common mistakes, Problem-solving strategies
