Algebra
Variables, Expressions, Linear Equations and Identities
---
Overview
Algebra forms the bridge between arithmetic and higher mathematics. In AP TET Paper I (Classes 1-5) and Paper II (Classes 6-8), algebra tests your ability to work with unknown quantities, manipulate expressions, and solve equations—skills essential for teaching mathematical reasoning to young learners.
For Paper I, expect basic introduction to variables and simple expressions. Paper II demands deeper understanding: forming and solving linear equations, applying algebraic identities, and recognising patterns. Questions typically involve direct computation, word problems requiring equation formation, and identity-based simplification.
Mastering algebra means understanding that letters represent numbers, expressions follow arithmetic rules, and equations are statements of equality that can be solved systematically. This conceptual clarity is what you must develop—and eventually teach.
---
Key Concepts
- **Variable**: A symbol (usually x, y, z) that represents an unknown or changeable quantity. Unlike constants (fixed numbers like 5 or -3), variables can take different values.
- **Algebraic Expression**: A combination of variables, constants, and operations (+, −, ×, ÷). Examples: 3x + 5, 2a² − 4b + 7. Expressions do NOT have an equals sign.
- **Terms, Coefficients, and Constants**: In 4x² + 3x − 7, there are three terms. The coefficient of x² is 4, coefficient of x is 3, and −7 is the constant term.
- **Like and Unlike Terms**: Like terms have identical variable parts (3x and 5x are like; 3x and 3x² are unlike). Only like terms can be combined.
- **Equation vs Expression**: An equation has an equals sign and states that two expressions are equal. 2x + 3 = 11 is an equation; 2x + 3 is an expression.
- **Linear Equation in One Variable**: An equation where the highest power of the variable is 1. Standard form: ax + b = 0, where a ≠ 0.
- **Solution/Root**: The value of the variable that makes the equation true. For 2x + 3 = 11, the solution is x = 4.
- **Algebraic Identity**: An equation true for ALL values of the variables involved, not just specific solutions.
---
Formulas / Key Facts
### Standard Algebraic Identities (Must Memorise)
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. **(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca** — Square of trinomial
5. **(x + a)(x + b) = x² + (a + b)x + ab** — Product of binomials with common variable
### Solving Linear Equations — Key Steps
- **Transposition Rule**: When a term moves across the equals sign, its sign changes. Addition becomes subtraction and vice versa; multiplication becomes division and vice versa.
- **Balance Method**: Whatever operation you perform on one side, perform on the other side to maintain equality.
### Important Facts
- Degree of a polynomial = highest power of the variable
- A linear expression has degree 1
- Coefficient of a term is the numerical part attached to the variable
- An equation can have one solution, no solution, or infinitely many solutions
---
Worked Examples
### Example 1: Simplifying Expressions **Simplify: 3x + 5y − 2x + 4 − 3y + 1**
Step 1: Group like terms
- x terms: 3x − 2x = x
- y terms: 5y − 3y = 2y
- Constants: 4 + 1 = 5
Step 2: Write simplified expression **Answer: x + 2y + 5**
---
### Example 2: Solving a Linear Equation **Solve: 3x − 7 = 2x + 5**
Step 1: Bring variable terms to left side 3x − 2x − 7 = 5 x − 7 = 5
Step 2: Bring constants to right side x = 5 + 7
**Answer: x = 12**
Verification: LHS = 3(12) − 7 = 36 − 7 = 29; RHS = 2(12) + 5 = 24 + 5 = 29 ✓
---
### Example 3: Using Algebraic Identity **Find the value of 103² using an identity.**
Step 1: Express 103 as (100 + 3)
Step 2: Apply (a + b)² = a² + 2ab + b² 103² = (100 + 3)² = 100² + 2(100)(3) + 3² = 10000 + 600 + 9
**Answer: 10609**
---
### Example 4: Word Problem **The sum of three consecutive integers is 72. Find the integers.**
Step 1: Let the integers be x, x + 1, x + 2
Step 2: Form equation x + (x + 1) + (x + 2) = 72 3x + 3 = 72
Step 3: Solve 3x = 69 x = 23
**Answer: 23, 24, 25**
---
Common Mistakes
- **Confusing expressions with equations** → Remember: expressions have no equals sign. You simplify expressions but solve equations.
- **Wrong sign during transposition** → When moving +5 to the other side, it becomes −5, not +5. Always reverse the sign.
- **Adding unlike terms** → Students write 3x + 2y = 5xy. This is wrong. Unlike terms cannot be combined; 3x + 2y stays as is.
- **Forgetting to apply identity to ALL terms** → In (a + b)², students write a² + b², forgetting the middle term 2ab. Always expand completely.
- **Arithmetic errors in multi-step equations** → Solve step by step. Write each step clearly. Verify your answer by substituting back.
- **Misidentifying coefficients** → In −5x, the coefficient is −5, not 5. The sign is part of the coefficient.
---
Quick Reference
- **Variable** = letter representing unknown; **Constant** = fixed number
- **Like terms**: same variable with same power → can be combined
- **(a + b)² = a² + 2ab + b²** — never forget the middle term
- **(a + b)(a − b) = a² − b²** — no middle term here
- To solve linear equations: isolate the variable using transposition or balance method
- Always verify solutions by substituting back into the original equation