Consider the following system of linear equations for four real variables $x_1,x_2,x_3,x_4$ :

$x_1-2x_3+x_4=5$

$3x_1+x_2-5x_3=8$

$x_1+2x_2-5x_4=-9$

$($a$)$ Write down the augmented matrix $(b|)$ of this system of linear equations.

$2$

$($b$)$ Bring the augmented matrix $(b|)$ into reduced row echelon form.

$4$

$($c$)$ Write down the solution set of the system of linear equations.

$4$

$($d$)$ Write down the solution set of the corresponding homogeneous system of linear equations.