The system of homogeneous simultaneous linear equations

a_{1}x + b_{1}y + c_{1}z = 0; a_{2}x + b_{2}y + c_{2}z=0; a_{3}x + b_{3}y + c_{3}z=0

has a non-trivial solution (i.e. at least one of x,y,z, is different from zero ) if

*Example -: If the system of equations 3x + 10 y + 17z = 0, –x + 6 y + 13 z = 0 and
20 x – 13y + *

*l*

*z = 0 has a non-trivial solution then find the solution.*** Solution:** The observation for A.P. property reveals l = -46 to have D = 0 (non-trivial solution)

Let z = k and by first two equations, we have

**3x +10 y = – 17 k **

–*3x + 18y = – 39 k, therefore y = – 2k and x = k*

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