In maths, we sometimes don't know the value of a number — or we want a rule that works for any number. We use letters like x, y, a, b, n to represent such numbers. These are called letter-numbers or variables.
Example: "Think of a number and add 5." If the number is n, we write: n + 5
Different people may think of different numbers — that is why we call it a variable (it can vary!). The answer changes depending on what number n stands for.
| Word Statement | Letter-Number Expression |
|---|---|
| A number increased by 7 | n + 7 |
| 3 times a number | 3n |
| Half of a number | n / 2 |
| A number decreased by 4 | n − 4 |
| Square of a number | n² |
| 5 more than twice a number | 2n + 5 |
🔄 Variable
A letter that can take different values.
Examples: x, y, a, b, n, m, p, q
x y n
🔒 Constant
A fixed number that never changes.
Examples: 3, 7, −5, 0, 100
3 7 −5
🔢 Coefficient
The number multiplied with a variable.
In 5x, coefficient is 5.
In −3y, coefficient is −3.
❓ Unknown
A variable whose value we want to find.
In "x + 5 = 8", x is the unknown.
x = 3
Coeff = 3, Var = x
| Expression | Variable | Constant | Coefficient of Variable |
|---|---|---|---|
| 4x + 7 | x | 7 | 4 |
| 3a − 2 | a | −2 | 3 |
| y + 10 | y | 10 | 1 (written as y, not 1y) |
| 5n | n | — | 5 |
| 8 | — | 8 | — |
When the coefficient is −1, we write −x, not −1x.
An algebraic expression is a combination of variables, constants and operations (+, −, ×, ÷).
Examples: 2x + 3 | 5y − 7 | a + b | 3n² | 4p − 2q + 1
| Word Statement | Expression |
|---|---|
| 7 more than x | x + 7 |
| 4 less than y | y − 4 |
| Product of 3 and n, plus 2 | 3n + 2 |
| 5 times a number minus 8 | 5n − 8 |
| Twice the sum of x and y | 2(x + y) |
| Square of a, decreased by 1 | a² − 1 |
Riya is x years old. Her brother is 3 years older.
Brother's age = x + 3
A box has n chocolates. 5 are eaten.
Remaining chocolates = n − 5
Monomial
Expression with 1 term
Examples: 3x, 7y, −5a, 4xy
Binomial
Expression with 2 terms
Examples: 2x + 1, 3a − b
Trinomial
Expression with 3 terms
Examples: x + y + 2, 2a − b + 3
Polynomial
Expression with one or more terms
All of the above are polynomials!
A term is a single part of an expression. Terms are separated by + or − signs.
Term 1 Term 2 Term 3 (constant)
| Term | Coefficient | Variable Part | Factors |
|---|---|---|---|
| 5x | 5 | x | 5, x |
| −3y | −3 | y | −3, y |
| 7ab | 7 | ab | 7, a, b |
| −xy | −1 | xy | −1, x, y |
| 4 | 4 | — (constant) | 4 |
✓ Like Terms
Same variable, same power
✗ Unlike Terms
Different variable or power
The coefficients can be different — only the variable parts need to be identical.
x-terms: 2x and 5x → LIKE (can be combined)
y-terms: 3y and −y → LIKE (can be combined)
Result: (2x + 5x) + (3y − y) = 7x + 2y
To add algebraic expressions: (1) Write the expressions, (2) Group like terms together, (3) Add the coefficients of like terms.
= (3x + 5x) + (2y − y)
= 8x + y
= (2a + a) + (3b − 2b) + (−1 + 4)
= 3a + b + 3
a − 2b + 4
3a + b + 3
= (4x² + x²) + (3x − 5x) + (2 + 1)
= 5x² − 2x + 3
When subtracting expressions, change the sign of EVERY term in the expression being subtracted, then add.
= 5x + 3y − 2x − y (change sign of 2x → −2x and y → −y)
= (5x − 2x) + (3y − y)
= 3x + 2y
= 4a − 2b + 1 − a − 3b + 4 (all signs of 2nd expression flipped)
= (4a − a) + (−2b − 3b) + (1 + 4)
= 3a − 5b + 5
a + 3b − 4
3a − 5b + 5
Wrong: (5x + 3) − (2x − 1) = 5x + 3 − 2x − 1 = 3x + 2 ✗
Correct: (5x + 3) − (2x − 1) = 5x + 3 − 2x + 1 = 3x + 4 ✓
The −1 inside the bracket becomes +1 because the subtraction flips its sign!
To find the value of an expression, we substitute (replace) the variable with a given number, then calculate.
= 3(4) + 5 = 12 + 5 = 17
= 2(3) − 7 = 6 − 7 = −1
= (3)² + 2(3) − 1 = 9 + 6 − 1 = 14
= 2 + 5 + (2)(5) = 2 + 5 + 10 = 17
| Expression | x = 1 | x = 2 | x = 3 | Pattern |
|---|---|---|---|---|
| 2x + 1 | 3 | 5 | 7 | Odd numbers! |
| x² | 1 | 4 | 9 | Perfect squares! |
| 3x − 1 | 2 | 5 | 8 | Add 3 each time |
| x(x + 1) | 2 | 6 | 12 | Rectangular numbers! |
One of the most powerful uses of letter-numbers is describing patterns with a single formula!
| Triangles (n) | Matchsticks used | Pattern |
|---|---|---|
| 1 | 3 | 2(1) + 1 |
| 2 | 5 | 2(2) + 1 |
| 3 | 7 | 2(3) + 1 |
| 4 | 9 | 2(4) + 1 |
| n | 2n + 1 | General formula! |
🪶 Matchstick Triangle Builder
1, 4, 7, 10… (add 3)
nth term = 3n − 2
n=1: 1, n=2: 4, n=3: 7 ✓
2, 4, 6, 8… (even)
nth term = 2n
n=1: 2, n=2: 4, n=3: 6 ✓
1, 4, 9, 16… (squares)
nth term = n²
n=1: 1, n=2: 4, n=3: 9 ✓
1, 3, 5, 7… (odd)
nth term = 2n − 1
n=1: 1, n=2: 3, n=3: 5 ✓
Formulas are expressions that describe relationships between real quantities. Variables in formulas stand for measurable things.
▢ Square
Perimeter = 4s
Area = s²
s = side length
▭ Rectangle
Perimeter = 2(l + b)
Area = lb
l = length, b = breadth
△ Triangle
Perimeter = a + b + c
a, b, c = three sides
🚗 Speed
Distance = S × T
Simple Interest = PRT/100
P=principal, R=rate, T=time
P = 2(l + b) = 2(6.5 + 4.2) = 2 × 10.7 = 21.4 cm
A = s² = 7² = 49 cm²
D = 60 × 3.5 = 210 km
SI = PRT/100 = (5000 × 8 × 3)/100 = 120000/100 = ₹1200