Step-by-step Solution:

The given system of equations is \[ x + y + 2z = a \] \[ x + z = b \] \[ 2x + y + 3z = c \] We solve for the condition under which this system has a solution. Step 1: Express Variables in Terms of \( x \) and \( z \) From the second equation, \[ y = a - x - 2z \] Substituting this into the third equation: \[ 2x + (a - x - 2z) + 3z = c \] \[ 2x + a - x - 2z + 3z = c \] \[ x + a + z = c \] Step 2: Express the Condition Using the second equation \( x + z = b \), we substitute it into \( x + a + z = c \): \[ b + a = c \] \[ c = a + b \] Conclusion: The system has a solution if \[ c = a + b \]