Algebra

Algebraic Expressions, Identities and Linear Equations

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Overview

Algebra forms the backbone of upper-primary mathematics and carries significant weight in WB TET Paper II. This topic bridges arithmetic and higher mathematics by introducing symbolic representation of numbers and relationships. Students transitioning from Classes 6–8 must master how to form, simplify and manipulate algebraic expressions before they can solve equations confidently.

For the WB TET exam, expect questions on identifying terms, coefficients and like terms; applying standard algebraic identities for quick computation; and solving linear equations in one and two variables. Pedagogy questions often ask how to introduce abstract algebraic concepts to young learners using concrete examples. A firm grasp here also supports later topics in geometry (coordinate geometry) and mensuration (formula manipulation).

Candidates should focus on the four standard identities, the distinction between expressions and equations, and systematic methods for solving linear equations—these appear most frequently.

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Key Concepts

• **Algebraic Expression**: A combination of constants, variables and operations (no equality sign). Example: 3x + 5y − 7.

• **Terms, Coefficients and Constants**: In 4x² − 3x + 2, the terms are 4x², −3x and 2; the coefficient of x² is 4; the constant term is 2.

• **Like and Unlike Terms**: Like terms have identical variable parts (3xy and −5xy); unlike terms differ in variables or powers (2x and 2x²).

• **Polynomial**: An expression with non-negative integer exponents. Classified by degree—linear (degree 1), quadratic (degree 2), cubic (degree 3).

• **Identity vs Equation**: An identity holds true for all values of variables; an equation is true only for specific values (the solutions).

• **Linear Equation in One Variable**: Form ax + b = 0 (a ≠ 0). Exactly one solution: x = −b/a.

• **Linear Equation in Two Variables**: Form ax + by + c = 0. Infinite solutions forming a straight line; a unique solution exists when paired with another such equation.

• **Solving Simultaneous Equations**: Use substitution or elimination to find the unique (x, y) satisfying both equations.

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Formulas / Key Facts

### Standard Algebraic Identities

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. 4. **(x + a)(x + b) = x² + (a + b)x + ab** — Product of two binomials with common variable.