If the system of linear equations x - 4 y + 7 z = g ; 3 y - 5 z = h ; - 2 x + 5 y - 9 z = k is consistent, then:

Step-by-step solution

x - 4 y + 7 z = g ...(i) 3 y - 5 z = h ...(ii) - 2 x + 5 y - 9 z = k ...(iii) (i) + (ii), we get x - y + 2 z = g + h ...(iv) (ii) - (iii), we get 2 x - 2 y + 4 z = h - k ⇒       x - y + 2 z = h - k 2 ...(v) Since, coefficient of x , y , z in (iv) and (v) are same. Hence, for system to be consistent g + h = h - k 2 ⇒       2 g + 2 h = h - k ⇒       2 g + h + k = 0