I just need to know if I am doing this right what I did was

I just need to know if I am doing this right, what I did was I took -2R1 + R2 --> R2 and came up with R1 - ( 1 4 -3 ) with Row 2 being (0 h-9 0) So it is consistent as long as h is not equal to 9, correct? Or is it inconsistent all the way?

Solution

The row operations which has been applied would result in second row as

< 0 , h - 8 , 0 >

If we take h = 8 which means the system of equations are as below

x₁ + 4x₂ = -3 ; 2x₁ + 8x₂ = -6

which shows that equations are linearly dependent and hence they consistently have infinite number of solutions .

Let the given system be as

a₁x + b₁y = c₁

a₂x + b₂y = c₂

where x , y are variable

The system would be finitely consistent if ( a₁/a₂ ) ( b₁/b₂ )

=> h 8

The system cannot be inconsistent as third entry is zero , if it had been a non-zero value for h = 8 , the system would have been inconsistent