Identities and Linear Equations

Overview

Algebraic identities and linear equations in two variables form the backbone of upper-primary algebra in the KAR TET Paper II Mathematics section. These topics bridge arithmetic and higher algebra, testing both computational fluency and conceptual understanding. Identities provide shortcuts for expanding and factorising expressions, while linear equations model real-world problems involving two unknown quantities.

For TET aspirants, this topic typically appears in two forms: direct questions testing identity application or factorisation, and word problems requiring formation and solution of simultaneous equations. Mastery here also supports pedagogy questions—you may be asked how to introduce these concepts to Class 8–10 students or identify common student errors.

Expect 2–4 questions from this combined topic. Focus on memorising standard identities, practising the three methods of solving linear pairs, and understanding graphical interpretation of solutions.

Key Concepts

• **Algebraic identity**: An equation true for all values of the variables involved. Unlike an equation (true for specific values), an identity holds universally. Example: (a + b)² = a² + 2ab + b² is always true.

• **Linear equation in two variables**: An equation of the form ax + by + c = 0, where a, b, c are real numbers and a, b are not both zero. The graph is always a straight line.

• **Solution of a linear equation**: An ordered pair (x, y) that satisfies the equation. A single linear equation has infinitely many solutions lying on its line.

• **Pair of linear equations**: Two linear equations in the same two variables. The solution is the point(s) where both equations are satisfied simultaneously.

• **Consistent system**: A pair with at least one solution. If exactly one solution exists, lines intersect (unique solution). If infinitely many solutions exist, lines coincide (dependent system).

• **Inconsistent system**: A pair with no solution—lines are parallel and distinct.

• **Condition ratios**: For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
• Unique solution: a₁/a₂ ≠ b₁/b₂
• No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
• Infinite solutions: a₁/a₂ = b₁/b₂ = c₁/c₂

Formulas / Key Facts

**Standard Algebraic Identities**

1. (a + b)² = a² + 2ab + b² 2. (a − b)² = a² − 2ab + b² 3. a² − b² = (a + b)(a − b) 4. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca 5. (a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b) 6. (a − b)³ = a³ − 3a²b + 3ab² − b³ = a³ − b³ − 3ab(a − b) 7. a³ + b³ = (a + b)(a² − ab + b²) 8. a³ − b³ = (a − b)(a² + ab + b²) 9. a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca) 10. If a + b + c = 0, then a³ + b³ + c³ = 3abc