**1) LIKE and UNLIKE TERMS in a linear expression**

In the expression, 2x+3y+4x+7y, all the terms in x and y are like terms. That is 2x and 4x are like terms just like 3y and 7y. Like terms only differ in their coefficients and at times their signs. An expression like 2x+3y+4x+7y can be simplified by combining like terms. Also, the terms 2x and 3y are unlike terms; we cannot simplify any expression with unlike terms because the variables are different. The term 2x= x+x or 2*x..

**2) SIMPLIFYING LINEAR EXPRESSIONS**

To simplify a linear expression, add or subtract like terms. Where there are no like terms, the expression is left the same.

**Example: **

Simplify as far as possible the following:

i) 8x+2-3x+4.

ii) 2x+3y-4x+2y.

iii) 3x-xy+2xy+5x.

iv) 2ab-3a-ab+5b.

**Solution**

i) 8x+2-3x+4= 5x+6.

ii) 2x+3y-4x+2y=-2x+5y.

iii) 3x-xy+2xy+5x=8x+xy.

iv) 2ab-3a-ab+5b=ab-3a+5b.

**3) SIMPLIFYING LINEAR EXPRESSIONS BY REMOVING BRACKETS**

Remove the brackets by multiplying all terms in the bracket by the one outside.

**Example: **

Simplify each of the following expressions;

i) 3(x+6).

ii) 4(a+b).

iii) 3(x+2)+5(x-3).

iv) 4(a+2)+3(2a-1).

v) 2(6n+1)-3(n-5).

vi) 3(2x+3)-2(y-2).

**Solution**

i) 3(x+6)= 3x+18.

ii) 4(a+b)=4a+4b.

iii) 3(x+2)+5(x-3)=3x+6+5x-15

3(x+2)+5(x-) =8x+9.

4(a+2)+3(2a-1)=4a+8+6a-3

4(a+2)+3(2a-1)=10a+5.

v) 2(6n+1)-3(a-5)=12n+2-3n+15

2(6n+1)-3(a-5)=9n+17.

Vi) 3(2x+3)-2(y-2)=6x+9-2y+4

3(2x+3)-2(y-2)=6x+13-2y.

**4) FACTORIZING LINEAR EXPRESSIONS**

Factorizing a linear expression is to write it as a product of two or more terms. This is done by removing the common factor(s) using the distributive law of multiplication and insert brackets.

**Example:**

Factorize each of the following simplifying your result as far as possible.

i) 3a+3b+3c

ii) ax+ay

iii) 2xy-4x

iv) 12a+18ab

v) 6pn-8xn

**Solution**

i) 3a+3b+3c=3(a+b+c).

ii) ax+ay =a(x+y).

iii) 2xy-4x=2.x.y-2.2.x (remove common factors)

2xy-4x=2x(y-2).

iv) 12a+18ab=6.2.a+6.3.a.b (remove common factors)

12a+18ab=6a(2+3b).

v) 6pn-8xn= 2.3.p.n-2.4.x.n (remove common factors)

6pn-8xn= 2n(3p-4x).

**5) EXPANDING LINEAR EXPRESSIONS**

This is done by removing the brackets found in the expression then

**Example:**

Expand each of the following simplifying your result as far as possible.

i) 3(2a+3b)

ii) (a+x)(b+y)

iii) (x+y)(x+y)

iv) (a+b)(a-b)

**Solution**

i) 3(2a+3b)=6a+9b.

ii) (a+x)(b+y)=ab+ay+bx+xy.

iii) (x+y)(x+y)=x²+xy+xy+y²

(x+y)(x+y)=x²+2xy+y².

iv) (a+b)(a-b)=a²-ab+ab-b²

(a+b)(a-b)=a²-b².