Simplifying The Algebraic Expression (12x + 10y)/2 + (12y - X)/5 A Step-by-Step Guide
In the realm of mathematics, simplifying complex algebraic expressions is a fundamental skill. It allows us to break down seemingly intricate problems into manageable components, making them easier to solve and understand. This article delves into the simplification and solution of the expression (12x + 10y)/2 + (12y - x)/5. We will explore the step-by-step process, highlighting key mathematical principles and techniques involved. Our goal is to provide a comprehensive guide that not only demonstrates the solution but also enhances your understanding of algebraic manipulation.
Understanding the Basics of Algebraic Expressions
Before we dive into the specific problem, let's first solidify our understanding of algebraic expressions. An algebraic expression is a combination of variables (like x and y), constants (numbers), and mathematical operations (+, -, ×, ÷). Simplifying these expressions involves applying the order of operations (PEMDAS/BODMAS) and using properties like the distributive property and combining like terms. These foundational concepts are crucial for tackling more complex problems.
Breaking Down the Expression (12x + 10y)/2 + (12y - x)/5
The expression we're addressing, (12x + 10y)/2 + (12y - x)/5, presents a blend of terms involving variables x and y, along with fractions. To simplify this, we'll first address each fraction individually and then combine the results. This approach allows us to manage the complexity in a structured manner. The first fraction, (12x + 10y)/2, can be simplified by dividing each term in the numerator by 2. This gives us 6x + 5y. The second fraction, (12y - x)/5, doesn't have an immediate common factor, so we'll keep it as is for now but remember that we'll need to address the denominator when combining terms.
Step-by-Step Simplification Process
1. Simplify Each Fraction Separately
Our first step is to simplify each fraction independently. For the first fraction, (12x + 10y)/2, we can divide both terms in the numerator by the denominator, 2. This yields:
(12x / 2) + (10y / 2) = 6x + 5y
This simplification makes the expression more manageable. For the second fraction, (12y - x)/5, there isn't an immediate simplification by division, so we'll leave it as is for now.
2. Finding a Common Denominator
To combine the simplified fractions, we need a common denominator. Since the denominators are 1 (for 6x + 5y) and 5 (for (12y - x)/5), the least common denominator (LCD) is 5. We'll multiply the first expression (6x + 5y) by 5/5 to get a common denominator:
(6x + 5y) * (5/5) = (30x + 25y) / 5
Now both parts of the expression have the same denominator, which allows us to combine them.
3. Combining the Fractions
Now that we have a common denominator, we can combine the two expressions:
(30x + 25y) / 5 + (12y - x) / 5
To combine them, we add the numerators and keep the common denominator:
(30x + 25y + 12y - x) / 5
4. Combining Like Terms
The next step involves combining like terms in the numerator. We have terms with 'x' and terms with 'y'. Let's group them together:
(30x - x) + (25y + 12y)
Combining these terms, we get:
29x + 37y
So, the expression becomes:
(29x + 37y) / 5
5. Final Simplified Expression
The simplified form of the expression (12x + 10y)/2 + (12y - x)/5 is:
(29x + 37y) / 5
This is the most simplified form we can achieve without additional information or constraints.
Alternative Approaches and Considerations
While the step-by-step method outlined above is straightforward, there are alternative approaches one could take. For instance, you could distribute the division across the terms in the numerator right from the beginning. However, this might lead to dealing with fractions earlier in the process. The key is to choose a method that you find most comfortable and that minimizes the chances of making errors.
Common Mistakes to Avoid
When simplifying algebraic expressions, there are some common mistakes to watch out for. One is forgetting to distribute the division across all terms in the numerator. Another is incorrectly combining like terms. Always double-check your work and ensure that you're applying the correct mathematical principles.
Real-World Applications
Simplifying algebraic expressions isn't just an academic exercise. It has numerous real-world applications in fields like physics, engineering, and computer science. For example, in physics, you might need to simplify equations to calculate the trajectory of a projectile or the current in an electrical circuit. In engineering, simplified expressions can help in designing structures and systems. Understanding these applications can make the learning process more engaging and meaningful.
Practice Problems
To solidify your understanding, here are a few practice problems you can try:
- Simplify: (15a + 20b)/5 + (9b - 2a)/3
- Simplify: (8x - 12y)/4 - (6y + x)/2
- Simplify: (21p + 14q)/7 + (10q - 3p)/5
Working through these problems will help you develop your skills and confidence in simplifying algebraic expressions.
In conclusion, simplifying the expression (12x + 10y)/2 + (12y - x)/5 involves a series of steps, including simplifying individual fractions, finding a common denominator, combining like terms, and arriving at the final simplified form: (29x + 37y) / 5. This process demonstrates the importance of understanding and applying basic algebraic principles. By mastering these techniques, you can tackle more complex mathematical problems and appreciate the elegance and power of algebra. Remember, practice is key to proficiency, so keep working on problems and expanding your mathematical toolkit.
