Variables and constants
A variable is a letter (x, y, n, a…) that stands for a number. Its value can change. A constant is a number whose value is fixed, like 5 or −2.
Why use letters? A letter lets us write one rule for many numbers. "A square has side s" gives perimeter 4s for every square: s = 3 cm gives 12 cm, s = 10 cm gives 40 cm.
An algebraic expression joins variables and constants with +, −, × or ÷. Examples: 3x + 2, 5a − 4b, x² + 7, n/2. An expression has no equals sign; an equation does (3x + 2 = 14).
Writing rules: 3 × x is written 3x (number first, no × sign); x × x is x²; 1x is just x.
Terms, coefficients and types of expressions
The parts of an expression joined by + or − are its terms. In 4x² − 3x + 7 the terms are 4x², −3x and 7. The sign in front belongs to the term.
- Coefficient: the number part of a term. In −3x it is −3. In x² it is 1.
- Constant term: a term with no letter (7).
- Factors of a term: 4x² = 4 × x × x.
Naming by number of terms
- Monomial: one term (5y)
- Binomial: two terms (3x + 2)
- Trinomial: three terms (x² + 2x + 1)
- Polynomial: one or more terms with whole-number powers
Reading the structure
Look at an expression as blocks. In 5(x + 2)², the whole (x + 2) is one block that is squared, then multiplied by 5. Seeing the blocks tells you what to do first and helps you spot, for example, that the value is never negative.
Like terms and simplifying
Like terms have exactly the same letters raised to the same powers: 3x and −5x; 2ab and 7ab; 4y² and y². Unlike terms differ: 3x and 3y; x and x²; x and 5.
To add or subtract like terms, add or subtract their coefficients and keep the letter part: 7x + 2x = 9x; 5ab − 8ab = −3ab. Unlike terms stay separate: 3x + 4 cannot become 7x.
Simplify = collect like terms. 2x + 3 + x + 1 = (2x + x) + (3 + 1) = 3x + 4. The two forms are equivalent: they give the same value for every x.
Multiplying terms
Multiply the numbers, then the letters: 3x × 4y = 12xy; 2a × 5a = 10a²; (−3m) × 2m² = −6m³.
Substitution: finding the value
To find the value of an expression, replace each letter by its number and work it out using the order of operations (brackets, powers, × and ÷, then + and −).
Example: 2a² − 3b when a = 3, b = −2: 2 × 3² − 3 × (−2) = 2 × 9 + 6 = 24.
A formula is an equation linking quantities: area of rectangle A = lw; speed v = d/t; simple interest I = PRT/100. Put numbers into the right side to get the left side. You can also rearrange a formula to make another letter the subject: from v = d/t we get d = vt.
Expanding brackets and factorising
Expanding uses the distributive law: a(b + c) = ab + ac. The outside term multiplies every term inside. 3(x + 2) = 3x + 6; −2(y − 5) = −2y + 10; x(x + 4) = x² + 4x.
In the 3D, 3(x + 2) is a rectangle 3 units tall and x + 2 long: its area is 3x + 6.
Factorising is the reverse: find the highest common factor of all terms and take it outside. 6x + 9 = 3(2x + 3); 4a² − 10a = 2a(2a − 5). Check by expanding back.
Identities
An identity is true for every value of the letters, for example 2(x + 3) = 2x + 6, or (a + b)² = a² + 2ab + b². An equation like 2x = 6 is true only for x = 3. Identities are studied in more depth in the lesson on algebraic identities.
Try it: tile algebra at home
Cut strips of paper for x (say 6 cm) and small squares for 1 (1 cm). Lay out 2x + 3 + x + 1, then push like pieces together. Count: 3 strips and 4 squares. Now lay 3 rows of "one strip + two squares" to see 3(x + 2) = 3x + 6. In the 3D free-play step, set a and b and predict the expanded form before you look.
Key formulas and definitions
- Term = coefficient × letter part, e.g. −3x² has coefficient −3
- Like terms: ax + bx = (a + b)x
- Distributive law: a(b + c) = ab + ac; a(b − c) = ab − ac
- Factorising: ab + ac = a(b + c) (take out the HCF)
- x × x = x²; xᵃ × xᵇ = xᵃ⁺ᵇ
- Identity: true for all values; equation: true for some values
Worked examples
1. Write the terms and the coefficient of y in 5y² − 7y + 4.
Terms: 5y², −7y, 4. Coefficient of y = −7 (the sign goes with it). Constant = 4.
2. Simplify 4a + 3b − a + 5b − 2.
Group like terms: (4a − a) + (3b + 5b) − 2 = 3a + 8b − 2.
3. Find the value of 3x + 2 when x = 4.
3 × 4 + 2 = 12 + 2 = 14.
4. Find the value of 2p² − 3q when p = −2 and q = 5.
p² = (−2)² = 4. So 2 × 4 − 3 × 5 = 8 − 15 = −7.
5. Expand and simplify 3(2x + 1) − 2(x − 4).
3(2x + 1) = 6x + 3; −2(x − 4) = −2x + 8. Add: 6x − 2x + 3 + 8 = 4x + 11.
6. Factorise 12m² − 18m.
HCF of 12 and 18 is 6; both terms have m. So 12m² − 18m = 6m(2m − 3). Check: 6m × 2m = 12m², 6m × 3 = 18m. ✓
7. A taxi charges a fixed ₹50 plus ₹12 per km. Write an expression for the fare for d km and find it for 15 km.
Fare = 50 + 12d. For d = 15: 50 + 180 = ₹230.
Common mistakes
- Adding unlike terms: 3x + 4 is NOT 7x. Only like terms can be joined.
- Forgetting the sign: in 5 − 2x the coefficient of x is −2, not 2.
- Multiplying only the first term: 3(x + 2) is 3x + 6, not 3x + 2.
- Squaring a negative wrongly: when x = −3, x² = 9 (not −9); use brackets (−3)².