Question

what is homogeneous linear equation &nonhomogeneous linear equation?what is consistence&incossistence?

Answer 1

A homogeneous system of linear equations is of the form below for two variables

$ax_1+bx_2=0\\ cx_1+dx_2=0\\$

i.e the values of the variables when substituted into any of the equation give zero.

However, non-homogeneous system of linear equations is of the form

$ax_1+bx_2=d_1\\ cx_1+dx_2=d_2$

where both $d_1,d_2$ are non-zero.

For three variables, homogeneous system of linear equations is of the form

$ax_1+bx_2+cx_3=0\\ dx_1+ex_2+fx_3=0\\ gx_1+hx_2+jx_3=0\\$

while a form of non-homogeneous system of linear equations is

$ax_1+bx_2+cx_3=d_1\\ dx_1+ex_2+fx_3=d_2\\ gx_1+hx_2+jx_3=d_3\\$

If any of the above systems of equations has a solution then such a system of equations is said to be consistent. If the system of equations does not have solution, then the system is inconsistent.

Consider the equations below

$x+y=5\\ -2x-2y=6\\$

x and y cannot cancel out due to the reoccurrence of the first equation in the second equation. i.e.

$-2(x+y)=6$

So it's inconsistent.

However, the equations below are consistent.

$x+y=5\\ -x+y=1\\$

Because x=2 and y=3.