1. **State the problem:** We have the system of equations: $$\begin{cases} 4x - 5y + 3z = -5 \\ 44x + 17y + kz = -40 \\ -8x - 8y + 3z = 5 \end{cases}$$ We want to find the values of $k$ for which the system is consistent. 2. **Recall the rule for consistency:** A system of linear equations is consistent if the equations do not contradict each other. For a system of three equations in three variables, this often means the coefficient matrix and augmented matrix must have the same rank. 3. **Form the coefficient matrix $A$ and augmented matrix $[A|b]$:** $$A = \begin{bmatrix} 4 & -5 & 3 \\ 44 & 17 & k \\ -8 & -8 & 3 \end{bmatrix}, \quad b = \begin{bmatrix} -5 \\ -40 \\ 5 \end{bmatrix}$$ 4. **Check for linear dependence:** If the second row is a linear combination of the first and third rows, the system might be inconsistent for some $k$. 5. **Use row operations or determinant:** Calculate the determinant of $A$: $$\det(A) = 4 \begin{vmatrix} 17 & k \\ -8 & 3 \end{vmatrix} - (-5) \begin{vmatrix} 44 & k \\ -8 & 3 \end{vmatrix} + 3 \begin{vmatrix} 44 & 17 \\ -8 & -8 \end{vmatrix}$$ Calculate each minor: $$\begin{aligned} M_1 &= 17 \times 3 - (-8) \times k = 51 + 8k \\ M_2 &= 44 \times 3 - (-8) \times k = 132 + 8k \\ M_3 &= 44 \times (-8) - 17 \times (-8) = -352 + 136 = -216 \end{aligned}$$ So, $$\det(A) = 4(51 + 8k) + 5(132 + 8k) + 3(-216) = 204 + 32k + 660 + 40k - 648 = (204 + 660 - 648) + (32k + 40k) = 216 + 72k$$ 6. **Set determinant not equal to zero for unique solution:** $$216 + 72k \neq 0 \implies 72k \neq -216 \implies k \neq -3$$ 7. **Conclusion:** The system is consistent for all $k$ except $k = -3$. **Final answer:** $$k \neq -3$$