If x = 6a + 8b + 9c; y = 2b - 3a - 6c and z = c - b + 3a; find :

(i) x + y + z

(ii) x - y + z

(iii) 2x - y - 3z

(iv) 3y - 2z - 5x

(i) If x = 6a + 8b + 9c; y = 2b - 3a - 6c and z = c - b + 3a, then

x + y + z = (6a + 8b + 9c) + (2b - 3a - 6c) + (c - b + 3a)

= 6a + 8b + 9c + 2b - 3a - 6c + c - b + 3a

Arrange the polynomial with like terms together and combine them.

= (6a - 3a + 3a) + (8b + 2b - b) + (9c - 6c + c)

= 6a + 9b + 4c

Hence, x + y + z = 6a + 9b + 4c

(ii) If x = 6a + 8b + 9c; y = 2b - 3a - 6c and z = c - b + 3a, then

x - y + z = (6a + 8b + 9c) - (2b - 3a - 6c) + (c - b + 3a)

= 6a + 8b + 9c - 2b + 3a + 6c + c - b + 3a

Arrange the polynomial with like terms together and combine them.

= (6a + 3a + 3a) + (8b - 2b - b) + (9c + 6c + c)

= 12a + 5b + 16c

Hence, x - y + z = 12a + 5b + 16c

(iii) If x = 6a + 8b + 9c; y = 2b - 3a - 6c and z = c - b + 3a, then

2x - y - 3z = 2(6a + 8b + 9c) - (2b - 3a - 6c) - 3(c - b + 3a)

= 12a + 16b + 18c - 2b + 3a + 6c - 3c + 3b - 9a

Arrange the polynomial with like terms together and combine them.

= (12a + 3a - 9a) + (16b - 2b + 3b) + (18c + 6c - 3c)

= 6a + 17b + 21c

Hence, 2x - y - 3z = 6a + 17b + 21c

(iv) If x = 6a + 8b + 9c; y = 2b - 3a - 6c and z = c - b + 3a, then

3y - 2z - 5x = 3(2b - 3a - 6c) - 2(c - b + 3a) - 5(6a + 8b + 9c)

= 6b - 9a - 18c - 2c + 2b - 6a - 30a - 40b - 45c

Arrange the polynomial with like terms together and combine them.

= (-9a - 6a - 30a) + (6b + 2b - 40b) + (-18c - 2c - 45c)

= -45a - 32b - 65c

Hence, 3y - 2z - 5x = -45a - 32b - 65c