Two institutions decided to award their employees for resourcefulness, competence and determination in the form of prizes at the rate of Rs. x, Rs. y, Rs. z respectively per person. The first institution decided to award respectively 4, 3 and 2 employees with total prize money Rs. 37,000 and second institution decided to award respectively 5, 3 and 4 employees with the total prize money of Rs. 47,000 for resourcefulness, competence and determination. All the prizes per person together amount to Rs. 12,000.
The linear equations representing the given information have
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Unique solution
The system is AX = B with
\( \displaystyle A = \begin{bmatrix}4 & 3 & 2\\ 5 & 3 & 4\\ 1 & 1 & 1\end{bmatrix},\quad X = \begin{bmatrix}x\\ y\\ z\end{bmatrix},\quad B = \begin{bmatrix}37000\\ 47000\\ 12000\end{bmatrix} \)
Expanding along the first row:
\( |A| = 4(3 - 4) - 3(5 - 4) + 2(5 - 3) \)
\( = -4 - 3 + 4 = -3 \)
Since |A| ≠ 0, A is invertible and \(X = A^{-1}B\) is the unique solution (x = 4000, y = 5000, z = 3000).