Definition of Multi-Step Algebraic Simplification for Grade 7

📚 Definition of Multi-Step Algebraic Simplification

Multi-step algebraic simplification is the process of reducing an algebraic expression to its simplest form by applying multiple mathematical operations in a specific order. This often involves combining like terms, using the distributive property, and applying the order of operations (PEMDAS/BODMAS) to eliminate parentheses and reduce the expression to its most concise form.

📜 History and Background

The roots of algebraic simplification can be traced back to ancient civilizations, including the Babylonians and Egyptians, who developed methods for solving linear and quadratic equations. However, the formalization of algebraic notation and simplification techniques evolved over centuries, with significant contributions from Greek, Arab, and Indian mathematicians. The development of symbolic algebra in the 16th and 17th centuries laid the foundation for the modern techniques used today.

🔑 Key Principles

• 🔢 Combining Like Terms: This involves adding or subtracting terms with the same variable and exponent. For example, $3x + 5x$ simplifies to $8x$.
• ➕ Addition and Subtraction: These operations are used to combine numerical terms and simplified variable terms.
• ✖️ Multiplication and Division: Applying the distributive property to remove parentheses and simplifying fractions. For example, $2(x+3)$ becomes $2x + 6$.
• ➗ Order of Operations (PEMDAS/BODMAS): Following the correct order of operations (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) is crucial for accurate simplification.
• ⚖️ Maintaining Equality: When simplifying equations, remember to perform the same operation on both sides to maintain balance.

🧮 Real-World Examples

Let's break down some examples:

Example 1:

Simplify: $3(x + 2) + 4x - 5$

1. Apply the distributive property: $3x + 6 + 4x - 5$
2. Combine like terms: $(3x + 4x) + (6 - 5)$
3. Simplify: $7x + 1$

Example 2:

Simplify: $2(2y - 1) - (y + 3)$

1. Apply the distributive property: $4y - 2 - y - 3$
2. Combine like terms: $(4y - y) + (-2 - 3)$
3. Simplify: $3y - 5$

📝 Practice Quiz

Simplify the following expressions:

1. $4(a - 3) + 2a$
2. $5b - 2(b + 1)$
3. $3(c + 2) - (c - 1)$
4. $2(3d - 4) + 5d$
5. $6e - 3(e - 2)$
6. $4(2f + 1) - 3f$
7. $5(g - 2) + 2g - 3$

✅ Solutions to Practice Quiz

1. $6a - 12$
2. $3b - 2$
3. $2c + 7$
4. $11d - 8$
5. $3e + 6$
6. $5f + 4$
7. $7g - 13$

⭐ Conclusion

Mastering multi-step algebraic simplification is crucial for success in algebra and beyond. By understanding the key principles and practicing regularly, you can develop the skills needed to confidently tackle more complex mathematical problems.