Juice Blend Prices and a Dishonest Row Operation
A canteen makes three juice blends, P, Q and R, from three ingredients. Each litre of P uses 2, 1 and 1 units of the ingredients; Q uses 1, 3 and 1; R uses 1, 1 and 2. The ingredient unit prices are x, y and z. On three consecutive days the canteen buys exactly the quantities needed for 1 litre of each blend, and the bills are ₹40, ₹54 and ₹38 respectively. The manager writes the system as A [x y z]^T = [40 54 38]^T, with A = [[2, 1, 1], [1, 3, 1], [1, 1, 2]]. A trainee, checking the accounts, replaces the third row by the sum of the first two rows and the third bill by ₹94, arguing that the new third equation is automatically true and therefore harmless. The manager is unsure whether unique prices can still be recovered and asks for a determinant-based decision before paying the supplier.
1. Solve the original system for x, y and z and verify by substitution. What does the value of det(A) tell you about the solution?
Read the worked answer
The system is:
2x + y + z = 40 x + 3y + z = 54 x + y + 2z = 38.
Subtract the first equation from the second: -x + 2y = 14, so x = 2y - 14. Subtract the first from the third: -x + z = -2, so z = x - 2. Substitute into the second: x + 3y + (x - 2) = 54, so 2x + 3y = 56. Using x = 2y - 14 gives 4y - 28 + 3y = 56, hence 7y = 84, y = 12. Then x = 10 and z = 8.
Check: 2(10) + 12 + 8 = 40; 10 + 3(12) + 8 = 54; 10 + 12 + 2(8) = 38.
Now det(A) = 2(6 - 1) - 1(2 - 1) + 1(1 - 3) = 10 - 1 - 2 = 7. Since det(A) ≠ 0, A is invertible and the system has a unique solution. The unique prices are ₹10, ₹12 and ₹8 per unit.
2. Analyse the trainee's modified system: row3 := row1 + row2 and bill3 := 94. Compute the new determinant, decide whether unique prices remain possible, and state what the ₹94 bill implies about consistency. Would the conclusion change if the third bill were ₹95?
Read the worked answer
The modified coefficient matrix has rows [2, 1, 1], [1, 3, 1] and [3, 4, 2]. The third row is exactly the sum of the first two rows, so the rows are linearly dependent and the determinant is 0. The first two rows are not multiples of each other, so the rank is 2. With 3 unknowns and rank 2, the system cannot have a unique solution.
Consistency depends on the right-hand side. Since row3 = row1 + row2, the third bill must equal 40 + 54 = ₹94. The trainee's value is exactly ₹94, so the modified system is consistent but has infinitely many solutions. Unique prices cannot be recovered from it.
If the third bill were ₹95, the left side would still force the third bill to be ₹94, so ₹95 would make the system inconsistent: no price vector could satisfy all three equations. In either case, the trainee's modification destroys uniqueness.