Simplification of algebraic expression

A combination of constants and variables connected by the signs of fundamental operations of addition, subtraction, multiplication and division is called an algebraic expression. That is, numbers or objects are represented with letters. For example, 5b + 3c + 7d can be representation of 5 babies, 3 cats and 7 dogs. The quantities 5b, 3c and 7d are referred to as the terms of the expression. The above expression contains 3 terms.

In the above expression, 5, 3 and 7 are called the co-efficient of b, c and d respectively.

Addition and Subtraction of Similar Terms

Examples:

Simplify the following expressions

(i) 4a + 5a + a  (ii) 9n – 5n – 3n  (iii) 11q + 5q – 2q + q  (iv) 20p – 9p – p + 2p

Solutions:

(i) 4a + 5a + a

Since the letters are alike, we add the co-efficients together

Therefore, 4a + 5a + a = 10a

(ii) 9n – 5n – 3n

Here, we have 9 n’s, 5 n’s, and 3 n’s

Therefore, 9n – 5n – 3n = 4n – 3n

= n

(iii) 11q + 5q – 2q + q

To simplify this type of expression, we group the positive and negative terms separately before finally simplifying it.

11q + 5q – 2q + q = 11q + 5q + q – 2q

= 16q – 2q

= 14q

(iv) 20p – 9p – p +2p = 20p + 2p – 9p – p

= 22p – 10p

= 12p

Multiplication and Division of Similar Terms

When we multiply algebraic expressions, the co-efficients multiply themselves and the unknown variables are written together. In division, the numerator and denominator is expressed in multiplication form and same variable divides each other.

Examples:

Simplify the following expressions

(i) 3 × a × b  (ii) 5y × 2ab  (iii) 6p × 3ap  (iv) 6y² ÷ 2y  (v) 36q ÷ 9q

Solutions:

(i) 3 × a × b = 3ab

(ii) 5y × 2ab = 5 × y × 2 × a b

= 5 × 2 × a × b × y

= 10aby

(iii) 6p × 3ap = 6 × p × 3 × a × p

= 6 × 3 × a × p × p

= 18 × a × p²

= 18ap²

(iv) 6y²÷ 2y = 63×y×y2×y

= 3 × y

= 3y

(v) 36q ÷ 9q = 364×q9×q

= 4 × 1

= 4

CLASS ACTIVITY

Simplify the following expressions;

(i) 20x – 4y – y – 3x

(ii) 8a + 14a – 21a

(iii) 3w × 2ab

(iv) 88p² ÷ 11p

(v) b – 7b + 3b + 8b + 2b

Collection and Simplification of Like and Unlike Terms in Algebraic Expressions

Algebraic terms such as 4a, 6a, -3a, −a6, a, etc are called like terms, because they all express terms of the same variable (a). On the other hand, algebraic terms such as 6u, 4w, 2y, 5x, 3z, etc. are called unlike terms, because each of them is expressed in terms of different variables u, w, y, x and z respectively.

NOTE: In simplifying algebraic expressions, like terms are grouped together.

Examples:

Simplify the following expressions:

(i) 4a + 5b – a – 4b

(ii) 18a – 3b – 6a + 10b

(iii) 6c – 10c – 7 + 13c

Solutions:

(i) 4a + 5b –a – 4b

= 4a – a + 5b – 4b

= 3a + b

(ii) 18a – 3a – 6a + 10b

= 9a + 10b

(iii) 6c – 10c – 7 + 13c

= 6c + 13c – 10c – 7

= 9c – 7

CLASS ACTIVITY

Simplify the following expressions:

(i) 9a + 10b – 5a – 4b

(ii) 7x + 5y – 4

(iii) 8h – 3 – 5h + 9

(iv) 4y – 2x + 5x – 3y

(v) 13a – b – 9a + 7b

(vi) 11x + 9y – 7x

The Use of Brackets

When brackets are used, the expression involving the brackets are worked out first.

Examples:

Simplify the following expressions:

(i) 2(b + 3a) – 2b + a

(ii) x( x + 2) + x² + 3x

(iii) 4(a + 3b) – 12b

Solutions:

(i) 2(b + 3a) – 2b

= 2b + 6a – 2b

= 2b – 2b + 6a

= 0 + 6a

= 6a

(ii) x(x + 2) + x² + 3a

= x² + 2x + x² + 3x

= x² + x² + 2x + 3x

= 2x² + 5x

(iii) 4(a + 3b) – 12b

= 4a + 12b – 12b

= 4a + 0

= 4a

CLASS ACTIVITY

Simplify the following expressions:

(a) 3(u + v) + 2(u – v)

(b) x(3x + 4) – 2x²

(c) 4(p + 2q) + 3(p + 2q)

PRACTICE QUESTIONS

Simplify the following expressions:

(i) 4xy² × 3x²y

(ii) 15g³h² ÷ 3gh²

(iii) 25w – 9z – 15w + 11z

(iv) 16q + 3(q- 2r)

(v) 5x + (3x + 4x)

ASSIGNMENT

Simplify the each of the following algebraic expressions

(i) 18m⁴n³ ÷ 6m²n²

(ii) 5x(3y – z )

(iii) 9ab x 3ab²

(iv) 3x – (2x – x) – (3y – 6y)

(v) (a – 3b) – (5a – 8b)

(vi) 6 – (9x – 7) + 2x

(vii) (3a + 5b) – (7a + 10b)

(viii) (14x + 5y) – (7x – 6y)