Question:

The given system of equations is:<br/>$x + 2y - 3z = p$<br/>$2x + 6y - 11z = q$<br/>$x - 2y + 7z = r$<br/>The determinant of the coefficients is:<br/>$D = \begin{vmatrix} 1 & 2 & -3 \\ 2 & 6 & -11 \\ 1 & -2 & 7 \end{vmatrix}$<br/>$= 1(42 - 22) - 2(14 + 11) - 3(-4 - 6)$<br/>$= 20 - 50 + 30 = 0$<br/>Now, $D_1 = \begin{vmatrix} p & 2 & -3 \\ q & 6 & -11 \\ r & -2 & 7 \end{vmatrix} = p(20) - 2(7q + 11r) - 3(-2q - 6r)$<br/>$= 20p - 14q - 22r + 6q + 18r$<br/>$= 20p - 8q - 4r = 4(5p - 2q - r)$<br/>If $D_1 = 0$, then there are infinite solutions which confirm at least one solution.<br/>$\therefore 5p - 2q - r = 0$