Problem regarding inconsistent solution 
My  attempt  :
For  First answer   i try  to used by elementary row operation $ A=  \begin{bmatrix} 1&2&3\\2&4&8 \\3&6&7 \end{bmatrix}$ that  give $ A=  \begin{bmatrix} 1&2&3\\0&0&2 \\0&0&0 \end{bmatrix} \begin{bmatrix} x_1 \\x_2\\x_3 \end{bmatrix}=\begin{bmatrix} 1 \\0\\5\end{bmatrix}$
so i got $ x_1 + 2x_2  +  3x_3=1$  , $2x_2=0$, $x_3= 5$
Now  i got $ x_1=-4, x_2=0 ,x_3 =5$
For  the  second  answer   it is  very easy  take  $b= (0,0,0)$  that will produce no solution
Is  its  true  ?

That way to proceed is not correct, indeed we need to apply the RREF on the augmented matrix
$$\left[\begin{array}{ccc|c} 1&2&3&1\\2&4&8&0 \\3&6&7&5 \end{array}\right]$$
For the second question let consider
$$\left[\begin{array}{ccc|c} 1&2&3&b_1\\2&4&8&b_2 \\3&6&7&b_3 \end{array}\right]$$
and apply RREF again to obtain
$$\left[\begin{array}{ccc|c} 1&2&3&b_1\\2&4&8&b_2 \\3&6&7&b_3 \end{array}\right]\to \left[\begin{array}{ccc|c} 1&2&3&b_1\\0&0&2&b_2-2b_1 \\0&0&-2&b_3-3b_1 \end{array}\right]\to \left[\begin{array}{ccc|c} 1&2&3&b_1\\0&0&2&b_2-2b_1 \\0&0&0&b_2+b_3-5b_1 \end{array}\right]$$
then a vector which leads to an inconsisten system is for example $(1,1,1)$.
