Algebra

Algebraic Expressions, Identities and Linear Equations

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Overview

Algebra forms the backbone of secondary mathematics and appears consistently in Assam TET Paper II. This topic tests your ability to manipulate symbols, simplify expressions, apply standard identities quickly, and solve equations—skills essential for any mathematics teacher working with classes VI to VIII.

For the exam, expect questions that combine multiple concepts: simplifying an expression using an identity, then solving for a variable. Mastery here also supports success in later topics like quadratic equations and geometry (where algebraic methods prove lengths or areas). Students who internalize the identities and practice systematic equation-solving will find this section high-scoring.

The key is pattern recognition. Once you see which identity applies or how to isolate the variable, the arithmetic becomes straightforward. Focus on speed and accuracy—errors in sign handling or transposition are the most common traps.

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Key Concepts

• **Algebraic Expression**: A combination of constants, variables and operations (e.g., 3x² + 2xy − 5). No equals sign—that makes it an equation.

• **Terms, Coefficients and Degree**: In 4x³y, the coefficient is 4, variables are x and y, and the degree is 3 + 1 = 4. Degree of a polynomial is the highest sum of exponents in any term.

• **Like and Unlike Terms**: Like terms share identical variable parts (3xy and −7xy). Only like terms can be added or subtracted directly.

• **Polynomial Classification**: Monomial (1 term), binomial (2 terms), trinomial (3 terms). Degree-based: linear (degree 1), quadratic (degree 2), cubic (degree 3).

• **Algebraic Identity**: An equation true for all values of variables. Identities enable quick expansion and factorization without long multiplication.

• **Linear Equation in One Variable**: Form ax + b = 0. Solution involves isolating x through inverse operations while maintaining equality.

• **Linear Equation in Two Variables**: Form ax + by + c = 0. Infinite solutions lying on a straight line; unique solution found when two such equations intersect.

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Formulas / Key Facts

### Standard Algebraic Identities

| Identity | Expanded Form | |----------|---------------| | (a + b)² | a² + 2ab + b² | | (a − b)² | a² − 2ab + b² | | (a + b)(a − b) | a² − b² | | (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) | | (a + b + c)² | a² + b² + c² + 2ab + 2bc + 2ca |