Introduction to Algebra
Algebra uses letters (variables) to represent unknown numbers, making it possible to solve general problems and write rules.
Why algebra?
- A rule for the perimeter of a square: P = 4 × s (works for any s)
- Instead of calculating for each side, one formula covers all cases.
Variables and Constants
Variable: A letter representing an unknown or changing value.
- Common variables: x, y, n, a, b, l, m, p, q
Constant: A fixed number that does not change.
- Examples: 5, 100, −3, ½
Example: Meena has n chocolates. If she gets 5 more, she has n + 5 chocolates.
- Here n is a variable (we don't know how many she started with).
- 5 is a constant.
Algebraic Expressions
An algebraic expression is a combination of variables, constants, and operations (+, −, ×, ÷).
| Expression | Parts |
|---|---|
| 3n | 3 × n |
| n + 7 | n plus 7 |
| 2x − 3 | 2 times x, minus 3 |
| 5y + 2z | 5 times y plus 2 times z |
Terms in an Expression
- 3x + 5 has two terms: 3x and 5
- 2a − b + 4 has three terms: 2a, −b, 4
Coefficient
The coefficient is the number multiplying the variable.
- In 7m, the coefficient of m is 7.
- In −3x, the coefficient of x is −3.
Writing Expressions from Word Problems
| Statement | Expression |
|---|---|
| n increased by 4 | n + 4 |
| n decreased by 7 | n − 7 |
| 5 times n | 5n |
| n divided by 3 | n/3 |
| 3 times n plus 2 | 3n + 2 |
| 2 less than n | n − 2 |
Equations
An equation is a mathematical statement that two expressions are equal, using an = sign.
Examples:
- n + 5 = 9 (is n = 4 the solution?)
- 3x = 15 (is x = 5 the solution?)
- 2y − 3 = 7 (what is y?)
Trial and Error Method
Try different values until the equation is satisfied.
Example: Solve n + 3 = 8.
| n | n + 3 | = 8? |
|---|---|---|
| 3 | 6 | No |
| 4 | 7 | No |
| 5 | 8 | Yes ✓ |
Solving Equations Systematically
The key principle: whatever you do to one side, do to the other.
Type 1: x + a = b → x = b − a
- n + 4 = 9 → n = 9 − 4 = 5
Type 2: x − a = b → x = b + a
- m − 3 = 7 → m = 7 + 3 = 10
Type 3: ax = b → x = b ÷ a
- 5p = 20 → p = 20 ÷ 5 = 4
Type 4: x/a = b → x = a × b
- y/3 = 6 → y = 6 × 3 = 18
Always verify your solution by substituting back.
Patterns and Rules Using Algebra
Algebra helps write general rules for patterns.
Example 1: Pattern in triangles (matchstick problem)
- 1 triangle = 3 matchsticks
- 2 triangles = 5 matchsticks
- 3 triangles = 7 matchsticks
- n triangles = 2n + 1 matchsticks
Example 2: Perimeter of a square with side s → P = 4s
Key Vocabulary
| Term | Meaning |
|---|---|
| Variable | Letter representing unknown/changing value |
| Constant | Fixed number |
| Expression | Combination of variables, constants, operations — no = sign |
| Equation | Two expressions set equal (has = sign) |
| Coefficient | Number multiplied by the variable |
| Solution | Value of variable satisfying the equation |
| Term | Each part of an expression separated by + or − |
Exam Tips
- Expression vs equation: equation has = sign; expression does not.
- Coefficient: in 5x, coefficient is 5; in x, coefficient is 1 (often forgotten).
- Always verify your solution by substituting back into the equation.
- "5 less than n" = n − 5 (NOT 5 − n) — word-to-expression translation is a common error.
- Matchstick pattern questions: find the rule and use it.
- The letter used for the variable doesn't matter: x, y, n, p are all the same concept.