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System of Homogeneous Linear Equations

A system of linear equations is said to be homogeneous if the sum of powers of variable in each term is one.
 
Let the three homogeneous linear equations in three unknown, x, y, and z be
 
a1x + b1y + c1z = 0 ...(4)
a2x + b2y + c2z = 0 ...(5)
a3x + b3y + c3z = 0 ...(6)
 
Clearly, x = 0, y = 0, and z = 0 is a solution of system of equations (4), (5), and (6). This solution is called a trivial solution. Any other solution is called a non-trivial solution. Now consider
 
Δ = 64790.png

Solutions under different conditions

  1. If Δ 0, then the given system of equations has only trivial solution and the number of solutions in this case is one.
  2. If Δ = 0, then the given system of equations has non-trivial solution as well as trivial solution and number of solutions in this case is infinite.




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