ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS

The addition and subtraction of algebraic expressions are almost similar to the addition and subtraction of numbers.

How to add and subtract algebraic expressions ?

When adding and subtracting algebraic expressions, we first categorize the terms into two types.

They are like terms and unlike terms.

Like terms :

The terms whose variables and exponents are the same are known as like terms. 

That means,

3x2 and 4x2 are like terms and they can be added or subtracted from each other.

When adding or subtracting like terms, the coefficients are added or subtracted and the variables remain unchanged.

We can only combine like terms by adding or subtracting them with one another.

Unlike terms :

The terms having different variables are unlike terms.

That means,

2x2 and 4y2 are unlike terms and they cannot be added or subtracted from each other.

Unlike terms cannot be combined by adding or subtracting them with one another.

Problem 1 :

Find the sum of the following expressions.

(i)  7p + 6q, 5p – q, q + 16p

(ii)  a + 5b + 7c, 2a + 10b + 9c

(iii)  mn + t, 2mn – 2t, -3t + 3mn

(iv)  u + v, u – v, 2u + 5v, 2u – 5v

(v)  5xyz – 3xy, 3zxy – 5yx

Solution :

(i) 

Given, 7p + 6q, 5p – q, q + 16p

= 7p + 6q + 5p – q + q + 16p

By combining like terms,

=  (7p + 5p + 16p) + (6q – q + q)

=  (7 + 5 + 16)p + (6 – 1 + 1)q

=  28p + 6q

(ii) 

Given, a + 5b + 7c, 2a + 10b + 9c

=  a + 5b+ 7c + 2a + 10b + 9c

By combining like terms,

=  (a + 2a) + (5b + 10b) + (7c + 9c)

=  (1 + 2)a + (5 + 10)b + (7 + 9)c

=  3a + 15b + 16c

(iii) 

Given, mn + t, 2mn – 2t, -3t + 3mn

= mn + t + 2mn – 2t - 3t + 3mn

By combining like terms,

=  (mn + 2mn + 3mn) + (t – 2t – 3t)

=  (1 + 2 + 3)mn + (1 – 2 – 3)t

=  6mn – 4t

(iv)

Given, u + v, u – v, 2u + 5v, 2u – 5v

=  u + v + u – v + 2u + 5v + 2u – 5v

By combining like terms,

=  (u + u + 2u + 2u) + (v – v + 5v – 5v)

= (1 + 1 + 2 + 2)u + (1 – 1 + 5 – 5)v

=  6u

(v)

Given, 5xyz – 3xy, 3zxy – 5yx

=  5xyz – 3xy + 3zxy – 5yx

By combining like terms,

=  (5xyz + 3zxy) – (3xy + 5yx)

=  (5 + 3)xyz – (3 + 5)xy

=  8xyz – 8xy

Problem 2 :

Subtract :

(i)  13x + 12y – 5 from 27x + 5y – 43

(ii)  3p + 5 from p – 2q + 7

(iii)  m + n from 3m – 7n

(iv)  2y + z from 6z – 5y

Solution :

(i)

Given,  13x + 12y – 5 from 27x + 5y – 43

 =  (27x + 5y – 43) – (13x + 12y – 5)

=  27x + 5y – 43 – 13x – 12y + 5

By combining like terms,

=  (27x – 13x) + (5y – 12y) – (43 – 5)

=  14x – 7y - 38

(ii)  

Given, 3p + 5 from p – 2q + 7

 =  (p – 2q + 7) - (3p + 5)

=  p - 2q + 7 - 3p - 5

=  -2p - 2q + 2

(iii)

Given, m + n from 3m – 7n

=  (3m – 7n) – (m + n)

=  3m – 7n - m - n

By combining like terms,

=  (3m – m) + (-7n - n)

=  (3 – 1)m + (-7 - 1)n

=  2m – 8n

(iv)

Given, 2y + z from 6z – 5y

=  (6z – 5y) – (2y + z)

=  (6z – 5y) + (-2y - z)

=  (6z - z) + (-5y – 2y)

=  (6 - 1)z + (-5 – 2)y

=  5z – 7y

Problem 3 :

Simplify :

(i)  (x + y – z) + (3x – 5y + 7z) – (14x + 7y – 6z)

(ii)  p + p + 2 + p + 3 – p – 4 – p – 5 + p + 10

(iii)  n + (m + 1) + (n + 2) + (m + 3) + (n + 4) + (m + 5)

Solution :

(i)

Given, (x + y – z) + (3x – 5y + 7z) – (14x + 7y – 6z)

By combining like terms,

=  (x + 3x – 14x) + (y – 5y – 7y) + (-z + 7z + 6z)

=  (1 + 3 – 14)x + (1 – 5 – 7)y + (-1 + 7 + 6)z

=  -10x – 11y + 12z

(ii)

Given, p + p + 2 + p + 3 – p – 4 – p – 5 + p + 10

By combining like terms,

=  (p + p + p – p – p + p) + (2 + 3 – 4 – 5 + 10)

=  (1 + 1 + 1 – 1 – 1 + 1)p + (6)

=  2p + 6

(iii)

Given, n + (m + 1) + (n + 2) + (m + 3) + (n + 4) + (m + 5)

By combining like terms,

=  (m + 1) + (m + 3) + (m + 5) + n + (n + 2) + (n + 4)

=  (m + m + m + 1 + 3 + 5 + n + n + n + 2 + 4)

=   3m + 3n + 15

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