Exercise 4.1.7

Multiply the 3 equations by $y=(1,1,-1)$. Then $x_1-x_2=1$ plus $x_2-x_3=1$ minus $x_1-x_3=1$ is $0=1$. Key point: This $y$ in $N\left(A^T\right)$ is not orthogonal to $b=(1,1,1)$ so $b$ is not in the column space and $Ax=b$ has no solution.