To solve the specific structural system \left\{\begin{array}{l} 3x - 4z = -1 \\ 2y + 3z = 2 \\ 2x + 3y = 6 \end{array} ight. by elimination, begin by addressing equations sharing variables to construct a two-variable problem. Multiplying the first equation by $3$ yields $9x - 12z = -3$, while multiplying the second equation by $4$ produces $8y + 12z = 8$. Adding these manipulated equations directly cancels out the $z$ term, creating the new two-variable equation $9x + 8y = 5$.

Form a secondary system by pairing this new structural equation with the original third equation: \left\{\begin{array}{l} 9x + 8y = 5 \\ 2x + 3y = 6 \end{array} ight.. Multiply the first equation by $-3$ to get $-27x - 24y = -15$ and the second equation by $8$ to deduce $16x + 24y = 48$. Adding these together eliminates $y$, isolating $-11x = 33$, which securely simplifies to $x = -3$. Substituting $x = -3$ back into the third original equation resolves the second coordinate as $y = 4$. Further substituting $y = 4$ into the second equation precisely reveals $z = -2$. Writing the solution as an ordered triple gives $(-3, 4, -2)$, which dependably verifies true across each base equation.