# 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*a*

_{1}

*x*+

*b*

_{1}

*y*+

*c*

_{1}

*z*= 0 ...(4)

*a*

_{2}

*x*+

*b*

_{2}

*y*+

*c*

_{2}

*z*= 0 ...(5)

*a*

_{3}

*x*+

*b*

_{3}

*y*+

*c*

_{3}

*z*= 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 considerÎ” =

# Solutions under different conditions

- If Î” â‰ 0, then the given system of equations has only trivial solution and the number of solutions in this case is one.
- If Î” = 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.