Simplify R^2 - 10r - R + 5r A Step By Step Guide

In the realm of mathematics, algebraic expressions serve as the building blocks for more complex equations and formulas. Simplifying these expressions is a fundamental skill that allows us to manipulate and solve mathematical problems efficiently. This article delves into the process of simplifying the expression r^2 - 10r - r + 5r, providing a step-by-step guide to effectively combine like terms and arrive at the most concise form.

Understanding Algebraic Expressions

Before we embark on the simplification journey, let's first establish a clear understanding of what algebraic expressions entail. An algebraic expression is a mathematical phrase that combines variables (symbols representing unknown values), constants (fixed numerical values), and mathematical operations (such as addition, subtraction, multiplication, and division). The expression r^2 - 10r - r + 5r perfectly exemplifies this definition, featuring the variable 'r', constants like 10 and 5, and operations like subtraction and addition.

Like Terms: The Key to Simplification

The concept of "like terms" is pivotal in the simplification process. Like terms are those that share the same variable(s) raised to the same power. In our expression, -10r, -r, and 5r are like terms because they all contain the variable 'r' raised to the power of 1. On the other hand, r^2 is not a like term with the others because it involves 'r' raised to the power of 2. The ability to identify like terms is crucial as it allows us to combine them, making the expression more concise.

Step-by-Step Simplification of r^2 - 10r - r + 5r

Now that we have laid the groundwork, let's proceed with the step-by-step simplification of the expression r^2 - 10r - r + 5r.

1. Identify Like Terms

The first step is to meticulously identify the like terms within the expression. As we discussed earlier, -10r, -r, and 5r are like terms, while r^2 stands alone.

2. Combine Like Terms

The next step involves combining the identified like terms. To do this, we simply add or subtract their coefficients (the numerical values in front of the variables). In our case, we have:

-10r - r + 5r = (-10 - 1 + 5)r = -6r

3. Write the Simplified Expression

Finally, we write the simplified expression by combining the result from step 2 with the term that was not a like term (r^2). This gives us:

r^2 - 6r

Therefore, the simplified form of the expression r^2 - 10r - r + 5r is r^2 - 6r.

Why Simplify Algebraic Expressions?

Simplifying algebraic expressions is not merely an exercise in mathematical manipulation; it serves several crucial purposes:

• Clarity and Conciseness: Simplified expressions are easier to understand and work with. They present the mathematical relationship in its most basic form, eliminating unnecessary clutter.
• Efficient Problem Solving: When solving equations or evaluating expressions, simplification reduces the number of steps required, saving time and effort.
• Foundation for Advanced Concepts: A solid understanding of simplification is essential for tackling more advanced algebraic concepts such as factoring, solving equations, and graphing functions.

Common Mistakes to Avoid

While the simplification process is relatively straightforward, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them:

• Combining Unlike Terms: This is perhaps the most frequent error. Remember, only like terms can be combined. Terms with different variables or different powers of the same variable cannot be added or subtracted.
• Incorrectly Applying the Distributive Property: The distributive property is used to multiply a term by a group of terms within parentheses. Errors can occur if the multiplication is not applied to every term inside the parentheses.
• Forgetting the Sign: When combining like terms, pay close attention to the signs (positive or negative) in front of the coefficients. A misplaced sign can lead to an incorrect result.
• Order of Operations (PEMDAS/BODMAS): Always follow the order of operations (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) when simplifying expressions. Failing to do so can result in errors.

Practice Makes Perfect

The key to mastering the simplification of algebraic expressions is consistent practice. Work through various examples, starting with simple ones and gradually progressing to more complex problems. The more you practice, the more comfortable and confident you will become in your ability to identify like terms, combine them accurately, and avoid common mistakes.

Examples and Practice Problems

To further solidify your understanding, let's work through a few more examples:

Example 1: Simplify 3x^2 + 2x - x^2 + 5x

• Identify like terms: 3x^2 and -x^2 are like terms; 2x and 5x are like terms.
• Combine like terms: 3x^2 - x^2 = 2x^2; 2x + 5x = 7x
• Simplified expression: 2x^2 + 7x

Example 2: Simplify 4y - 7 + 2y + 3 - y

• Identify like terms: 4y, 2y, and -y are like terms; -7 and 3 are like terms.
• Combine like terms: 4y + 2y - y = 5y; -7 + 3 = -4
• Simplified expression: 5y - 4

Practice Problems:

1. Simplify 5a + 3b - 2a + b
2. Simplify 2p^2 - 4p + 5p^2 - p + 3
3. Simplify -3m + 6n - 2m - 4n + 1

Real-World Applications of Simplifying Algebraic Expressions

Simplifying algebraic expressions is not just a theoretical exercise confined to textbooks and classrooms. It has practical applications in various real-world scenarios. Here are a few examples:

• Calculating Costs: Imagine you are buying items at a store. You can use algebraic expressions to represent the total cost, where variables represent the price of each item and constants represent fixed costs like taxes. Simplifying the expression allows you to easily calculate the total cost for different quantities of items.
• Mixing Solutions: In chemistry and other fields, you might need to mix solutions of different concentrations. Algebraic expressions can represent the amount of solute in each solution, and simplifying the expression helps you determine the final concentration of the mixture.
• Designing Structures: Architects and engineers use algebraic expressions to model the forces and stresses on structures. Simplifying these expressions allows them to optimize designs and ensure stability.
• Computer Programming: Simplifying algebraic expressions is crucial in computer programming, where variables represent data and expressions represent calculations. Efficiently simplified expressions lead to faster and more reliable programs.

Conclusion: Mastering the Art of Simplification

Simplifying algebraic expressions is a fundamental skill in mathematics that empowers you to manipulate and solve problems effectively. By understanding the concept of like terms and following the step-by-step process outlined in this article, you can confidently simplify a wide range of expressions. Remember, practice is key to mastery, so work through plenty of examples and don't hesitate to seek help when needed. With a solid grasp of simplification, you'll be well-equipped to tackle more advanced mathematical concepts and real-world applications.

By consistently applying the principles discussed, you can master the art of simplification and unlock a deeper understanding of algebra and its applications.

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