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):
Isolate the variable by performing the same operation on both sides
Addition/subtraction to move constants; multiplication/division to remove coefficients
Always verify by substituting the solution back
**Degree of a Polynomial**:
Monomial: count sum of all exponents (3x²y³ has degree 5)
**Forgetting to distribute negative signs**: In 3 - (2x - 5), students write 3 - 2x - 5 instead of 3 - 2x + 5. → When removing brackets preceded by minus, change signs of ALL terms inside.
**Confusing identities**: Mixing up (a - b)² = a² - 2ab + b² with a² - b². → Remember: square of a binomial always has THREE terms; difference of squares has TWO terms.
**Adding unlike terms**: Writing 3x + 5x² = 8x³ is wrong. → Only like terms (same variable with same power) can be combined: 3x + 5x² stays as is.
**Sign errors when solving equations**: Moving +5 from LHS to RHS and keeping it +5. → When a term crosses the equals sign, its sign reverses: +5 becomes -5.
**Incomplete verification**: Solving correctly but not checking the answer when time permits. → Substituting back catches careless errors, especially in MCQs where one mark matters.
Quick Reference
**Expression**: no equals sign; **Equation**: has equals sign
**Linear equation**: variable power is 1; solve by isolating the variable
**(a + b)² = a² + 2ab + b²** — middle term is POSITIVE with plus sign
**(a - b)² = a² - 2ab + b²** — middle term is NEGATIVE with minus sign
**a² - b² = (a + b)(a - b)** — difference of squares, product of sum and difference