Algebra

Variables, Expressions, Linear Equations and Identities

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Overview

Algebra forms the bridge between arithmetic and higher mathematics. In AP TET Paper I (Classes 1-5) and Paper II (Classes 6-8), algebra tests your ability to work with unknown quantities, manipulate expressions, and solve equations—skills essential for teaching mathematical reasoning to young learners.

For Paper I, expect basic introduction to variables and simple expressions. Paper II demands deeper understanding: forming and solving linear equations, applying algebraic identities, and recognising patterns. Questions typically involve direct computation, word problems requiring equation formation, and identity-based simplification.

Mastering algebra means understanding that letters represent numbers, expressions follow arithmetic rules, and equations are statements of equality that can be solved systematically. This conceptual clarity is what you must develop—and eventually teach.

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Key Concepts

• **Variable**: A symbol (usually x, y, z) that represents an unknown or changeable quantity. Unlike constants (fixed numbers like 5 or -3), variables can take different values.

• **Algebraic Expression**: A combination of variables, constants, and operations (+, −, ×, ÷). Examples: 3x + 5, 2a² − 4b + 7. Expressions do NOT have an equals sign.

• **Terms, Coefficients, and Constants**: In 4x² + 3x − 7, there are three terms. The coefficient of x² is 4, coefficient of x is 3, and −7 is the constant term.

• **Like and Unlike Terms**: Like terms have identical variable parts (3x and 5x are like; 3x and 3x² are unlike). Only like terms can be combined.

• **Equation vs Expression**: An equation has an equals sign and states that two expressions are equal. 2x + 3 = 11 is an equation; 2x + 3 is an expression.

• **Linear Equation in One Variable**: An equation where the highest power of the variable is 1. Standard form: ax + b = 0, where a ≠ 0.

• **Solution/Root**: The value of the variable that makes the equation true. For 2x + 3 = 11, the solution is x = 4.

• **Algebraic Identity**: An equation true for ALL values of the variables involved, not just specific solutions.

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Formulas / Key Facts

### Standard Algebraic Identities (Must Memorise)

1. **(a + b)² = a² + 2ab + b²** — Square of a sum

2. **(a − b)² = a² − 2ab + b²** — Square of a difference

3. **(a + b)(a − b) = a² − b²** — Difference of squares