Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
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.