To solve the system $\left\{\begin{array}{l} x + 2y - z = 1 \\ 2x + 7y + 4z = 11 \\ x + 3y + z = 4 \end{array}\right.$ by elimination, first eliminate the variable $x$. Multiplying the first equation by $-1$ and adding it to the third equation yields the two-variable equation $y + 2z = 3$. Next, multiply the first equation by $-2$ and add it to the second equation to generate another two-variable equation, $3y + 6z = 9$. This forms a new sub-system: $\left\{\begin{array}{l} y + 2z = 3 \\ 3y + 6z = 9 \end{array}\right.$. To eliminate a variable from this new sub-system, multiply the first equation by $-3$ and add it to the second equation. This completely eliminates the variables, resulting precisely in the inherently true mathematical statement $0 = 0$. Because the resulting statement is true, the system is explicitly dependent and has infinitely many solutions. To determine the comprehensive solution set, express two variables precisely in terms of the third. Solving $y + 2z = 3$ for $y$ gives $y = -2z + 3$. Substituting this expression for $y$ into the first equation ($x + 2y - z = 1$) and solving strategically for $x$ yields $x = 5z - 5$. The complete solution is definitively any ordered triple logically of the form $(5z - 5, -2z + 3, z)$, where $z$ is uniquely any real number.