1. **Problem Statement:** Find all values of $k$ for which the augmented matrix \(\begin{bmatrix} 1 & k & -4 \\ 4 & 8 & 2 \end{bmatrix}\) corresponds to a consistent linear system. 2. **Understanding Consistency:** A system of linear equations is consistent if it has at least one solution. For two equations, this means the equations are not contradictory. 3. **Write the system from the matrix:** $$ \begin{cases} 1 \cdot x_1 + k \cdot x_2 = -4 \\ 4 \cdot x_1 + 8 \cdot x_2 = 2 \end{cases} $$ 4. **Use elimination or substitution:** Multiply the first equation by 4: $$ 4x_1 + 4k x_2 = -16 $$ 5. **Subtract the second equation from this:** $$ (4x_1 + 4k x_2) - (4x_1 + 8x_2) = -16 - 2 $$ Simplifies to: $$ 4k x_2 - 8 x_2 = -18 $$ $$ (4k - 8) x_2 = -18 $$ 6. **Analyze for consistency:** - If $4k - 8 = 0$, then the left side is zero, so the equation becomes $0 = -18$, which is impossible (no solution). - If $4k - 8 \neq 0$, then we can solve for $x_2$ and the system is consistent. 7. **Solve for $k$ when inconsistent:** $$ 4k - 8 = 0 \implies 4k = 8 \implies k = 2 $$ 8. **Conclusion:** - The system is consistent for all $k$ except $k = 2$. **Final answer:** $$ \boxed{k \neq 2} $$