The system of equations

x + y + 2z = a

x + z = b

2x + y + 3z = c

has a solution if

Given system of equations:
$x + y + 2z = a$
$x + z = b$
$2x + y + 3z = c$
Writing in matrix form $[A|B]$:
$\left[\begin{array}{ccc|c} 1 & 1 & 2 & a \\ 1 & 0 & 1 & b \\ 2 & 1 & 3 & c \end{array}\right]$
Applying $R_2 \rightarrow R_2 - R_1$ and $R_3 \rightarrow R_3 - 2R_1$:
$\left[\begin{array}{ccc|c} 1 & 1 & 2 & a \\ 0 & -1 & -1 & b-a \\ 0 & -1 & -1 & c-2a \end{array}\right]$
Applying $R_3 \rightarrow R_3 - R_2$:
$\left[\begin{array}{ccc|c} 1 & 1 & 2 & a \\ 0 & -1 & -1 & b-a \\ 0 & 0 & 0 & c-2a-(b-a) \end{array}\right]$
$\left[\begin{array}{ccc|c} 1 & 1 & 2 & a \\ 0 & -1 & -1 & b-a \\ 0 & 0 & 0 & c-a-b \end{array}\right]$
For system to have a solution:
$\rho(A) = \rho(A|B)$
$\Rightarrow c - a - b = 0$
$\therefore \boxed{a + b = c}$