Algebra: Variables, Expressions, Linear Equations and Identities

Overview

Algebra forms the foundation of mathematical reasoning and is a crucial section in GTET Mathematics. This topic bridges arithmetic and higher mathematics, testing your ability to work with unknown quantities, form and solve equations, and apply standard identities. For TET-1 (classes 1-5), expect basic introduction to variables and simple expressions. For TET-2 (classes 6-8), questions go deeper into linear equations, identities, and their applications.

Mastery of algebra is essential because it appears directly in the content section and indirectly supports problem-solving in arithmetic, mensuration, and data handling. Exam questions typically test your ability to simplify expressions, solve equations, and expand or factorise using identities. Speed and accuracy come from understanding concepts thoroughly rather than memorising procedures blindly.

Key Concepts

• **Variable**: A symbol (usually x, y, z) representing an unknown or changing quantity. Constants are fixed values (like 5, -3, π).

• **Algebraic Expression**: A combination of variables, constants, and operations. Example: 3x + 5y - 7 is an expression with three terms.

• **Terms, Coefficients, and Like Terms**: In 4x²y, the coefficient is 4. Like terms have identical variable parts (3xy and -5xy are like terms; 3xy and 3x²y are not).

• **Polynomial**: An expression with one or more terms where variables have whole number exponents. Classified by number of terms (monomial, binomial, trinomial) and by degree (highest power of variable).

• **Equation vs Expression**: An expression has no equality sign; an equation states two expressions are equal. Example: 2x + 3 is an expression; 2x + 3 = 7 is an equation.

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

• **Identity**: An equation true for all values of the variable(s). Example: (a + b)² = a² + 2ab + b² holds for every a and b.

Formulas / Key Facts

**Standard Algebraic Identities** (memorise these — they appear frequently):

| Identity | Expanded Form | |----------|---------------| | (a + b)² | a² + 2ab + b² | | (a - b)² | a² - 2ab + b² | | (a + b)(a - b) | a² - b² | | (x + a)(x + b) | x² + (a + b)x + ab | | (a + b + c)² | a² + b² + c² + 2ab + 2bc + 2ca | | (a + b)³ | a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b) | | (a - b)³ | a³ - 3a²b + 3ab² - b³ = a³ - b³ - 3ab(a - b) | | a³ + b³ | (a + b)(a² - ab + b²) | | a³ - b³ | (a - b)(a² + ab + b²) |

**Solving Linear Equations** (one variable):