Question

If the system of equations $2x-3y+5z=12,3x+y+pz=q$ and $x-7y+8z=17$ is consistent, then which of the following is not true?

• A. $p=2,q=7$
• B. $p\neq 2,q=7$
• C. $p\neq 2,q\neq 7$
• D. $p=2,q\neq 7$

Answer 1

Answer: (A)

Using Cramer’s rule:
$\Delta =\begin{vmatrix} 2 & -3 & 5 \\ 3 & 1 & p \\ 1 & -7 & 8 \end{vmatrix}=2\left(8 + 7 p\right)+3\left(24 - p\right)+5\left(- 21 - 1\right)$
$=-22+11p=11\left(p - 2\right)$
$\left(\Delta \right)_{1}=\begin{vmatrix} 12 & -3 & 5 \\ q & 1 & p \\ 17 & -7 & 8 \end{vmatrix}=12\left(8 + 7 p\right)+3\left(8 q - 17 p\right)+5\left(- 7 q - 17\right)$
$=33p-11q+11=11\left(3 p - q + 1\right)$
$\left(\Delta \right)_{2}=\begin{vmatrix} 2 & 12 & 5 \\ 3 & q & p \\ 1 & 17 & 8 \end{vmatrix}=2\left(8 q - 17 p\right)-12\left(24 - p\right)+5\left(51 - q\right)$
$=11\left(q - 2 p - 3\right)$
$\left(\Delta \right)_{3}=\begin{vmatrix} 2 & -3 & 12 \\ 3 & 1 & q \\ 1 & -7 & 17 \end{vmatrix}=2\left(17 + 7 q\right)+3\left(51 - q\right)+12\left(- 21 - 1\right)$
$=11q-77=11\left(q - 7\right)$
For unique solution $ \rightarrow p\neq 2,q\in R$
For infinite solutions $ \rightarrow p=2,q=7$