The price of 3 commodities X,Y,z are rupees x,y,z per unit respectively. A purchase 4 units of z and sells 3 units of X and 5 units of Y. B purchases 3 units of Y and sells 2 units of X and 1 unit of Z. C purchases 1 unit of X and sells 4 units of Y and 6 units of Z. In this process A, B and C earn Rs.6,000, Rs5,000 and Rs.13,000 respectively. Find the price per unit of 3 commodities by using Cramer's rule.
In this numerical the number associated with earning and selling are considered positive while purchasing which means outflow of cash is considered negative.
Equation for A:
3x + 5y - 4z = 6000 ....(i)
Equation for B:
2x - 3y + z = 5000 ....(ii)
Equation for C:
-x + 4y + 6z = 13000 ....(iii)
D =
3
2
-1
5
-3
4
-4
1
6
= 3 × (-3) × 6 + 5 × 1 × (-1) + (-4) × 2 × 4 - 3 × 1 × 4 - 5 × 2 × 6 - (-4) × (-3) × (-1)
= -54-5-32-12-60+12
= -151
D₁ =
6000
5000
13000
5
-3
4
-4
1
6
= 6000 × (-3) × 6 + 5 × 1 × 13000 + (-4) × 5000 × 4 - 6000 × 1 × 4 - 5 × 5000 × 6 - (-4) × (-3) × 13000
= -108000 + 65000-80000-24000-150000-156000
= -453000
D₂ =
3
2
-1
6000
5000
13000
-4
1
6
= 3 × 5000 × 6 + 6000 × 1 × (-1) + (-4) × 2 × 13000 - 3 × 1 × 13000 - 6000 × 2 × 6 - (-4) × 5000 × (-1)
= 90000-6000-104000-39000-72000-20000
= -151000
D₃ =
3
2
-1
5
-3
4
6000
5000
13000
= 3 × (-3) × 13000 + 5 × 5000 × (-1) + 6000 × 2 × 4 - 3 × 5000 × 4 - 5 × 2 × 13000 - 6000 × (-3) × (-1)
= -117000-25000 + 48000-60000-130000-18000
= -302000
x =
D₁
D =
-453000
-151 = 3000
y =
D₂
D =
-151000
-151 = 1000
z =
D₃
D =
-302000
-151 = 2000
D₁
D =
-453000
-151 = 3000
y =
D₂
D =
-151000
-151 = 1000
z =
D₃
D =
-302000
-151 = 2000
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