Suppose I have a system of equations whose augmented matrix is $$ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 7\\ 4 & 5 & 6 & 8 \end{array}\right] $$ After converting this matrix to row reduced echelon form I got $$ \left[\begin{array}{ccc|c} 1 & 0 & -1 & -19/3 \\ 0 & 1 & 2 & 20/3 \end{array}\right] $$
Now I want a solution such that $$0 < x_1 < 9, \quad 0 < x_2 < 7, \quad 0 < x_3 < 2$$ One such solutions is $x_1 = -16/3 , x_2 = 14/3$ and $x_3 = 1$.
But how do I find one such solution in general case, suppose I have n unknowns variables ? I am only considering the case when infinite solutions exist for a linear system