Algebra (TGT) — Study Notes for HP TET
Overview
Algebra forms the bridge between arithmetic and higher mathematics, introducing students to the power of using symbols to represent unknown quantities. For the HP TET Mathematics section, algebra questions typically focus on two core areas: manipulating algebraic expressions and solving linear equations. These concepts appear directly in content-based questions and also in pedagogy questions that ask how to teach algebraic thinking to elementary and middle-school learners.
Mastering algebra for this exam means being comfortable with forming expressions from word problems, simplifying expressions using basic rules, and solving equations systematically. The questions are usually straightforward but require careful attention to signs and the order of operations. Since HP TET assesses your ability to teach, understanding why algebraic methods work is as important as getting the right answer.
Key Concepts
- Variable: A letter (usually x, y, z) that represents an unknown or changeable quantity. It is not a fixed number but a placeholder.
- Constant: A fixed numerical value that does not change (e.g., 5, −3, π).
- Algebraic Expression: A combination of variables, constants, and operations (e.g., 3x + 5, 2a² − 4a + 7). Expressions do not have an equals sign.
- Term: Each part of an expression separated by + or − signs. In 4x² − 3x + 2, there are three terms: 4x², −3x, and 2.
- Coefficient: The numerical factor attached to a variable. In 7y, the coefficient is 7.
- Like Terms: Terms with the same variable raised to the same power. 5x and −2x are like terms; 5x and 5x² are not.
- Linear Equation: An equation where the highest power of the variable is 1 (e.g., 2x + 3 = 11). The graph is a straight line.
- Solution of an Equation: The value of the variable that makes the equation true. For 2x + 3 = 11, the solution is x = 4.
Formulas / Key Facts
| Concept | Formula / Rule | Context |
|---|---|---|
| Addition of like terms | ax + bx = (a + b)x | Combine coefficients only |
| Subtraction of like terms | ax − bx = (a − b)x | Watch the signs carefully |
| Distributive property | a(b + c) = ab + ac | Used to expand brackets |
| Solving linear equation | If ax + b = c, then x = (c − b)/a | Isolate variable step by step |
| Transposition rule | Moving a term across = changes its sign | +5 becomes −5 when moved |
| Identity: (a + b)² | a² + 2ab + b² | Square of a binomial sum |
| Identity: (a − b)² | a² − 2ab + b² | Square of a binomial difference |
| Identity: (a + b)(a − b) | a² − b² | Difference of squares |
| Linear equation in two variables | ax + by = c | Forms a straight line on graph |
Standard form of linear equation in one variable: ax + b = 0, where a ≠ 0.
Worked Examples
Example 1: Simplify the expression
Problem: Simplify 5x − 3 + 2x + 7 − x
Solution:
- Group like terms: (5x + 2x − x) + (−3 + 7)
- Combine variable terms: 5 + 2 − 1 = 6, so 6x
- Combine constants: −3 + 7 = 4
- Answer: 6x + 4
Example 2: Solve a linear equation
Problem: Solve for x: 3x − 7 = 14
Solution:
- Add 7 to both sides: 3x − 7 + 7 = 14 + 7
- Simplify: 3x = 21
- Divide both sides by 3: x = 21 ÷ 3
- Answer: x = 7
Verification: 3(7) − 7 = 21 − 7 = 14 ✓
Example 3: Equation with variable on both sides
Problem: Solve: 5x + 3 = 2x + 12
Solution:
- Bring variable terms to one side: 5x − 2x = 12 − 3
- Simplify: 3x = 9
- Divide by 3: x = 3
- Answer: x = 3
Verification: LHS = 5(3) + 3 = 18; RHS = 2(3) + 12 = 18 ✓
Example 4: Word problem
Problem: The sum of a number and 8 is 15. Find the number.
Solution:
- Let the number be x
- Form equation: x + 8 = 15
- Solve: x = 15 − 8 = 7
- Answer: The number is 7
Example 5: Using algebraic identity
Problem: Find the value of 103² using an identity.
Solution:
- Write 103 as (100 + 3)
- Apply (a + b)² = a² + 2ab + b²
- = 100² + 2(100)(3) + 3²
- = 10000 + 600 + 9
- Answer: 10609
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Adding unlike terms: 3x + 2 = 5x | Unlike terms cannot be combined. 3x + 2 stays as 3x + 2. |
| Forgetting to change sign when transposing: x + 5 = 12 → x = 12 + 5 | When moving +5 to RHS, it becomes −5. So x = 12 − 5 = 7. |
| Confusing expression with equation | An expression has no = sign (e.g., 2x + 3). An equation has = and can be solved. |
| Applying distributive property incorrectly: 2(x + 3) = 2x + 3 | Multiply BOTH terms inside: 2(x + 3) = 2x + 6. |
| Dropping the negative sign: −(x − 4) = −x − 4 | Distribute the minus: −(x − 4) = −x + 4. |
| Dividing only one side of the equation | Whatever operation you do to one side, do the same to the other side. |
Quick Reference
- Like terms: Same variable, same power — combine them; unlike terms cannot be added.
- Transposition: Moving a term across the = sign reverses its sign (+ becomes −, × becomes ÷).
- Linear equation: Highest power of variable is 1; exactly one solution.
- Always verify: Substitute your answer back into the original equation to check.
- Distributive property: a(b + c) = ab + ac — essential for expanding brackets.
- Identity shortcuts: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; (a + b)(a − b) = a² − b².