Simplifying Expressions With Distributive Property 20-5(9m+7)
In the realm of mathematics, simplifying expressions is a fundamental skill. This skill allows us to take complex mathematical statements and reduce them to their most basic, understandable forms. One of the most powerful tools in our arsenal for this task is the distributive property. The distributive property allows us to multiply a single term by multiple terms within parentheses, effectively removing the parentheses and paving the way for further simplification. In this article, we will delve deep into the application of the distributive property by simplifying the expression 20 - 5(9m + 7), providing a step-by-step guide that will enhance your understanding and proficiency in this crucial mathematical concept.
Understanding the Distributive Property
Before we dive into the specifics of our expression, let's first solidify our understanding of the distributive property itself. At its core, the distributive property states that for any numbers a, b, and c, the following equation holds true:
a( b + c ) = a b + a c
In simpler terms, this means that when you multiply a number ( a ) by a sum ( b + c ), you can distribute the multiplication across each term within the sum. You multiply a by b, then a by c, and finally add the results together. This seemingly simple property is the key to unlocking and simplifying a wide range of algebraic expressions. Mastering this property is crucial for success in algebra and beyond, as it forms the foundation for more advanced mathematical concepts and problem-solving techniques. The distributive property isn't just a mathematical trick; it's a fundamental concept that reflects the way multiplication interacts with addition and subtraction. It allows us to break down complex expressions into smaller, more manageable parts, making them easier to understand and manipulate.
Step-by-Step Simplification of 20 - 5(9m + 7)
Now that we have a firm grasp of the distributive property, let's apply it to the expression 20 - 5(9m + 7). We'll break down the simplification process into clear, manageable steps:
Step 1: Distribute the -5
The first, and often most critical, step is to correctly distribute the -5 across the terms within the parentheses. It's crucial to remember that we are distributing a negative number, which will affect the signs of the terms inside the parentheses. This is where many students make mistakes, so pay close attention to the signs.
-5 * (9m) = -45m
-5 * (7) = -35
By multiplying -5 by both 9m and 7, we've successfully applied the distributive property. Remember to treat the minus sign in front of the 5 as part of the coefficient, making it a negative 5. This is a common area for errors, so double-checking this step is always a good idea.
Step 2: Rewrite the Expression
After distributing, we rewrite the expression, replacing the parentheses with the results of our multiplication:
20 - 5(9m + 7) becomes 20 - 45m - 35
Notice how the parentheses have disappeared, and we now have a simpler-looking expression. The expression is now a series of terms that can be combined, which brings us to the next step.
Step 3: Combine Like Terms
In this step, we identify and combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, the like terms are the constant terms: 20 and -35. The term -45m is the only term with the variable 'm', so it will remain as is.
Combining the constant terms, we get:
20 - 35 = -15
Step 4: Final Simplified Expression
Now, we rewrite the expression with the combined like terms:
-45m - 15
This is the completely simplified form of the original expression, 20 - 5(9m + 7). We have successfully applied the distributive property and combined like terms to arrive at our final answer.
Common Mistakes to Avoid
When simplifying expressions using the distributive property, several common mistakes can occur. Being aware of these pitfalls can help you avoid them and ensure accurate results.
- Forgetting the Negative Sign: A frequent error is failing to distribute the negative sign along with the number. For instance, in our example, distributing -5 instead of just 5 is crucial. Always pay close attention to the sign preceding the parentheses.
- Incorrect Multiplication: Simple multiplication errors can derail the entire process. Double-check your multiplication facts, especially when dealing with larger numbers or negative signs.
- Combining Unlike Terms: Only like terms can be combined. Terms with different variables or exponents cannot be added or subtracted. Make sure you are only combining terms that have the exact same variable part.
- Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). Distribution should occur before addition or subtraction. If you perform operations in the wrong order, you will likely arrive at an incorrect answer.
Practice Makes Perfect
The best way to master the distributive property is through practice. Work through various examples, gradually increasing the complexity of the expressions. Start with simpler problems and then move on to those involving multiple variables, exponents, and nested parentheses.
Here are a few practice problems you can try:
- Simplify: 3(2x + 5)
- Simplify: -2(4y - 3)
- Simplify: 7 - (3z + 2)
- Simplify: 4(a + b) - 2(a - b)
By working through these and other practice problems, you'll build confidence and proficiency in using the distributive property.
Conclusion
The distributive property is a powerful tool in the world of algebra. By understanding and applying this property correctly, you can simplify complex expressions and solve a wide range of mathematical problems. Remember to distribute carefully, paying attention to signs, and combine like terms accurately. With practice and attention to detail, you'll master this essential skill and unlock new levels of mathematical understanding. In this article, we have shown in detail how to simplify the expression 20 - 5(9m + 7), providing a step-by-step guide and highlighting common mistakes to avoid. Keep practicing, and you'll be simplifying expressions like a pro in no time!
Remember, the key to success in mathematics is not just memorizing rules and formulas, but truly understanding the underlying concepts. The distributive property is more than just a trick; it's a fundamental principle that governs how multiplication interacts with addition and subtraction. By grasping this concept, you'll be well-equipped to tackle more advanced mathematical challenges in the future.
