Linear Programming and Matrix Reducing to Row Echelon Form

1. The Copperfield Mining Company owns two mines, each of which produces three grades of ore-high, medium, and low. The company has s contract to supply a smelting company with at least 12 tons of high-grade ore, 8 tons of medium-grade ore, and 24 tons of low-grade ore. Each mine produces a certain amount of each type of ore during each hour that it operates. Mine 1 produces 6 tons of high grade ore, 2 tons of medium-grade ore, and 4 tons of low-grade ore per hour. Mine 2 produces 2, 2, and 12 tons, respectively, or high-, medium-, and low-grade ore per hour. It cost Copperfield $200 per hour to mine each ton of ore from mine 1, and $160 per hour it needs to operate each mine so that its contractual obligations can be met at the lowest cost.

a. Formulate a linear programming model for this problem.
b. Solve this model using graphical analysis.

2. A clothier makes coats and slacks. The two resources required are wool cloth and labor. The clothier has 150 square yards of wool and 200 hours of labor available. Each coat requires 3 yards of wool and 10 hours of labor, whereas each pair of slacks requires 5 square yards of wool and 4 hours of labor. The profit for a coat is $50, and the profit for slacks is $40. The clothier wants to determine the number of coats and pairs of slacks to make so that profit will be maximized.

a. Formulate a linear programming model for this problem.
b. Solve this model using graphical analysis.

3. Solve this system of equations using Matrix. Show all Matrixes and reduce to row echelon form. Do not use decimals, use fractions.

2x-3y + z - w = -8
x + y - z - w = -4
x + y + z + w = 22
x - y - z - w = -14

4. Solve this system of equation using Matrix. Show all Matrixes and reduce to row echelon form. Do not use decimals, use fractions.

5x + y - z = 7
2x + 5y + 2z = 0
3x + y + z = 11