If the system of equations
\[
\begin{cases}
2x + y + pz = -1 \\
3x - 2y + z = q \\
5x - 8y + 9z = 5
\end{cases}
\]
has more than one solution, then \( q - p \) is equal to:
Show Hint
- \(2\)
- \(-2\)
- \(4\)
- \(-4\)
The Correct Option is C
Solution and Explanation
For a system of three linear equations to have more than one solution, the determinant of the coefficient matrix must be zero and the system must be consistent.
Step 1: Determinant of coefficient matrix \[ \begin{vmatrix} 2 & 1 & p\\ 3 & -2 & 1\\ 5 & -8 & 9 \end{vmatrix} =2((-2)\cdot 9-1\cdot(-8)) -1(3\cdot 9-1\cdot 5) +p(3\cdot(-8)-(-2)\cdot 5) \] \[ =2(-18+8)-1(27-5)+p(-24+10) =-20-22-14p \] \[ \Rightarrow -42-14p=0 \Rightarrow p=-3 \] Step 2: Consistency condition With \(p=-3\), observe that the rows satisfy: \[ -2R_1+3R_2=R_3 \] For consistency, the constants must satisfy the same relation: \[ -2(-1)+3q=5 \Rightarrow 2+3q=5 \Rightarrow q=1 \] Step 3: Compute \(q-p\) \[ q-p=1-(-3)=4 \] \[ \boxed{4} \]
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