Algebra
Algebraic Expressions, Identities and Linear Equations
Overview
Algebra forms the bridge between arithmetic and higher mathematics, making it a crucial topic for MAHA TET Paper II. This section tests your ability to manipulate symbols, simplify expressions, apply standard identities and solve equations — skills essential for teaching upper-primary students who encounter formal algebra for the first time in Classes VI–VIII.
Expect questions that assess both content knowledge and pedagogical understanding. You must know how to form and simplify algebraic expressions, recall and apply the four standard identities, and solve linear equations in one or two variables. Questions often combine these skills — for instance, using an identity to simplify an expression before solving an equation.
Mastering this topic also strengthens your preparation for geometry (where algebraic methods appear in mensuration formulas) and data handling (where expressions describe relationships). Allocate sufficient practice time here; the concepts reappear across the mathematics curriculum.
Key Concepts
- Algebraic Expression: A combination of constants, variables and operations (e.g., 3x + 5, 2a² − 7ab + 4). It does not contain an equality sign.
- Terms, Coefficients and Like Terms: Each part separated by + or − is a term. The numerical factor is the coefficient. Like terms share the same variable part (e.g., 5xy and −2xy are like terms).
- Monomial, Binomial, Trinomial, Polynomial: Classified by number of terms — one term (monomial), two terms (binomial), three terms (trinomial), many terms (polynomial).
- Degree of an Expression: The highest sum of exponents of variables in any term. For 4x³y², the degree is 3 + 2 = 5.
- Algebraic Identity: An equation true for all values of the variables. Differs from an equation, which is true only for specific values.
- Linear Equation: An equation where the highest power of the variable is 1. Standard form in one variable: ax + b = 0 (a ≠ 0).
- Solution / Root: The value of the variable that satisfies the equation.
- Transposition Rule: When a term moves across the equality sign, its sign changes (equivalent to adding/subtracting on both sides).
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 |
| (x + a)(x + b) | x² + (a + b)x + ab |
Useful Derived Results
- a² + b² = (a + b)² − 2ab = (a − b)² + 2ab
- (a + b)² − (a − b)² = 4ab
- (a + b)² + (a − b)² = 2(a² + b²)
Linear Equations
- One variable: ax + b = 0 → x = −b/a
- Two variables: ax + by + c = 0 represents a straight line; solution is an ordered pair (x, y).
- Simultaneous equations solved by substitution or elimination method.
Worked Examples
Example 1 — Simplifying an Expression
Simplify: 3(2x − 5) + 4(x + 2) − 7x
Solution:
- Expand brackets: 6x − 15 + 4x + 8 − 7x
- Group like terms: (6x + 4x − 7x) + (−15 + 8)
- Simplify: 3x − 7
Answer: 3x − 7
Example 2 — Applying an Identity
Evaluate 103² using an identity.
Solution: Write 103 as (100 + 3). Apply (a + b)² = a² + 2ab + b²
103² = (100 + 3)² = 100² + 2 × 100 × 3 + 3² = 10000 + 600 + 9 = 10609
Answer: 10609
Example 3 — Solving a Linear Equation
Solve: 5x − 3 = 2x + 9
Solution:
- Transpose 2x to left side: 5x − 2x − 3 = 9
- Simplify: 3x − 3 = 9
- Transpose −3: 3x = 9 + 3 = 12
- Divide both sides by 3: x = 4
Verification: LHS = 5(4) − 3 = 17; RHS = 2(4) + 9 = 17 ✓
Answer: x = 4
Example 4 — Simultaneous Equations (Elimination)
Solve: 2x + 3y = 12 and 4x − 3y = 6
Solution: Add the two equations to eliminate y: (2x + 3y) + (4x − 3y) = 12 + 6 6x = 18 → x = 3
Substitute x = 3 in first equation: 2(3) + 3y = 12 → 6 + 3y = 12 → 3y = 6 → y = 2
Answer: x = 3, y = 2
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Confusing expression with equation — attempting to "solve" 3x + 5. | An expression has no equality sign; you can only simplify it, not solve it. |
| Forgetting to change sign when transposing (e.g., moving +5 as +5 instead of −5). | Transposing is shorthand for adding/subtracting on both sides; sign must flip. |
| Applying (a + b)² as a² + b² and omitting the middle term 2ab. | Always expand fully: a² + 2ab + b². Drill the four identities until automatic. |
| Adding unlike terms (e.g., 2x + 3y written as 5xy). | Only like terms (same variable part) can be combined. |
| Multiplying only the first term inside a bracket by the factor outside. | Distribute to every term: a(b + c) = ab + ac. |
Quick Reference
- Expression → no "="; Equation → has "=" and can be solved.
- (a + b)² = a² + 2ab + b² — never forget the middle term.
- (a − b)² = a² − 2ab + b² — middle term is negative.
- (a + b)(a − b) = a² − b² — difference of squares; middle terms cancel.
- Transposition rule: term crosses "=" → sign flips.
- Verify solutions by substituting back into the original equation.
Keep these identities on fingertips and practise forming equations from word problems — both are high-frequency exam areas.