To solve the linear system \left\{\begin{array}{l} 4x - 3z = -5 \\ 3y + 2z = 7 \\ 3x + 4y = 6 \end{array} ight. using the elimination method, systematically pair equations to reduce the problem's dimensionality. Eliminate $z$ from the first two equations by multiplying the first by $2$ (yielding $8x - 6z = -10$) and the second by $3$ (yielding $9y + 6z = 21$). Integrating these formulas cancels $z$ to reliably state a new equation: $8x + 9y = 11$.

Bringing this calculated statement together with the unused third equation outlines a localized two-variable system: \left\{\begin{array}{l} 8x + 9y = 11 \\ 3x + 4y = 6 \end{array} ight.. Next, choose to eliminate $y$ by multiplying the first equation by $4$ to obtain $32x + 36y = 44$ and the second by $-9$ to create $-27x - 36y = -54$. Adjoining these equations strictly isolates $5x = -10$, simplifying cleanly to $x = -2$. Substituting $x = -2$ directly back into the third original equation allows deduction of the localized variable $y$ as $3$. Furthermore, placing $y = 3$ into the second equation predictably reveals $z = -1$. When written compactly, the functional solution successfully manifests as the tested ordered triple $(-2, 3, -1)$.