If the system of equations $x - 2y + 5z = 3$, $2x - y + z = 1$, and $11z - 7y + pz = q$ has infinitely many solutions, then
Step-by-Step Solution
Key Concept: For a system to have infinite solutions, all four determinants $D$, $D_1$, $D_2$, and $D_3$ must equal zero simultaneously.
For infinite solutions, we need $D = 0$, $D_1 = 0$, $D_2 = 0$, and $D_3 = 0$. Computing $D = \begin{vmatrix} -1 & 2 & 5 \\ 2 & -1 & 1 \\ 1 & 7 & p \end{vmatrix} = 0$ gives $-p + 7 + 4p - 22 - 15 = 0$, so $p = 10$. Computing $D_1$ with the condition gives $q = 8$. Computing $D_2$ with the condition also gives $q = 8$. Computing $D_3$ with the condition again gives $q = 8$. Therefore $p = 10$ and $q = 8$, so $p - q = 2$.
Correct Answer: 2