Question

Discuss the nature of solutions of the following system of equations 

(i) x + 2y – z = 6; -3x – 2y + 5z = -12; x – 2z = 3 

(ii) 2y + z = 3 (-x + 1); -x + 3y -z = -4; 3x + 2y + z = – 1/2

(iii) (y + z)/4 = (z + x)/3 = (x + y)/2; x + y + z = 27

Answer 1

(i) x + 2y – z = 6 … (1) 

-3x – 2y + 5z = -12 … (2) 

x – 2z = 3 … (3)

We see that the system has an infinite number of solutions.

(ii) 2y + z = 3(-x + 1);

-x + 3y – z = -4; 

3x + 2y + z = – 1/2

2y + z + 3x = 3 ⇒ 3x + 2y + z = 3 ... (1)

-x + 3y – z = -4 … (2) 

3x + 2y + z = – 1/2 … (3)

This is a contradiction. This means the system is inconsistent and has no solutions.

(iii) (y + z)/4 = (z + x)/3 = (x + y)/2; x + y + z = 27

Sub. x = 3 in (4) ⇒ 5(3) – z = 0 

15 – z = 0 

-z = -15 

z = 15

Sub, x = 3, z = 15 in (3) 

x + y + z = 27 

3 + y + 15 = 27 

y = 27 – 18 = 9 

x = 3, y = 9, z = 15 

∴ The system has unique solutions.