Solving and Simplifying Algebraic Expressions: A Comprehensive Guide

Algebraic expressions are a fundamental part of mathematics. They provide a way to express relationships between variables and numbers. In this guide, we'll explore how to solve and simplify algebraic expressions, with a focus on expressions involving variables like a, b, m, and n. If you're studying algebra or need to brush up on your skills, this article will be invaluable.

Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. For example, the expression 3a 6b 5 - 2a - b involves the variables a and b. In this article, we'll delve into how to simplify and solve such expressions by combining like terms.

Simplifying Expressions Involving Variables a and b

When you encounter an expression like 3a 6b 5 - 2a - b, the first step is to simplify it. You can do this by combining like terms. Here's a step-by-step guide:

Identify the like terms. In this case, we have two terms involving 'a' (3a and -2a) and two terms involving 'b' (6b and -b). Add or subtract like terms. Combine 3a and -2a to get 1a (or simply a), and combine 6b and -b to get 5b. Include any constant terms in the final expression. Here, the constant term is 5.

Therefore, the simplified expression is a 5b 5.

Simplifying Expressions Involving Variables m and n

Lets now consider the expression m 5n - m - 3n. Again, follow the same steps:

Identify the like terms. We have one term involving 'm' (m and -m) and two terms involving 'n' (5n and -3n). Combine like terms. Adding m and -m gives 0. Combining 5n and -3n gives 2n. No constant terms are present, so the final simplified expression is 2n.

Expansion and Multiplication in Algebra

In addition to simplification, another common task in algebra is expansion and multiplication. This involves distributing a term over a set of terms enclosed in parentheses.

Example: Expanding (2a 3)(a - 1)

Let's break down the process step-by-step:

Distribute 2a over (a - 1). This gives 2a^2 - 2a. Distribute 3 over (a - 1). This gives 3a - 3. Combine the results: 2a^2 - 2a 3a - 3. Combine like terms: 2a^2 a - 3.

The final expanded expression is 2a^2 a - 3.

Conclusion

Solving and simplifying algebraic expressions involves a series of steps, including identifying and combining like terms, and perhaps expanding or simplifying expressions through multiplication. Understanding and mastering these techniques is crucial for success in algebra and higher-level mathematics.

Key Takeaways

Identify and combine like terms to simplify expressions. Use the distributive property to expand expressions. Solve expressions by performing the necessary operations, such as addition, subtraction, and multiplication.

By practicing these techniques, you'll become more proficient in working with algebraic expressions. To further improve your skills, consider solving a variety of problems and referring to algebra textbooks or online resources.

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