Sarthaks Test
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If a,b,c are non – zero real numbers and if the system of equations :

(a – 1) x = y + z 

(b – 1) y = z + x 

(c – 1) z = x + y 

Has a non – trivial solution, 

then prove that -

ab + bc + ca = abc.

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Given Equations :

(a – 1) x = y + z 

(b – 1) y = z + x 

(c – 1) z = x + y 

Rearranging these equations,

(a – 1)x – y – z = 0 

– x + (b – 1)y – z = 0 

– x – y + (c – 1)z = 0 

For trivial solution, 

D = 0

= (a – 1)((b – 1)(c – 1) – (– 1)×(– 1)) + 1(– 1(c – 1) – (– 1)×(– 1)) – 1((– 1)×(– 1) + (b – 1))

= (a – 1)(bc –b –c + 1 – 1) + (1 – c – 1) – 1(1 + b – 1)) 

= (a – 1)(bc –b –c) –c –b 

= abc –ab –ac –bc + b + c – b – c 

= abc –ab –ac –bc

Now,

D = 0 

⇒ abc – ab – ac – bc = 0 

⇒ abc = ab + bc + ac 

Hence proved.

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