Linear Equations
- The system of
homogeneous simultaneous linear equations
- has a non-trivial
solution (i.e. at least one of x,y,z, is different from zero ) if
- If D >
0, then the only solution of the above system of equations is x = 0, y = 0, z =
0 (called trivial solution). .
- If D = 0, then the number of solutions of
the system of equations is infinite (called non-trivial and includes trivial
solution).
Example.1
If the system of
equations 3x + 10 y + 17z = 0, x + 6y + 13z = 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- trival 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
Example.2
If a, b, c are in G.P. with common ratio r1
and a , b and g are also in G.P. with common ratio r2 then
find the condition that r1 and r2 must satisfy in order
that the system of equations ax+ ay+z=
0, bx +by+z= 0, cx + gy +z = 0 has only a trivial
solution.
Solution
As a, b, c are in G.
P. with common ratio r1 and a,
b, g are in G. P. having common ratio r2,
a > 0, a >
0, b= ar1, c= ar12, b = ar2, g
= ar22.
Also the system of
equation have only zero( trivial ) solution
D = 
=
aa 
= aa ( r1 - 1) (r2
- 1) ( r1 - r2) >
0
&⇒ r1 > 1, r2 > 1 and r1 > r2 ( as a, a > 0 ). .
