I would like a detailed explanation of the following augmented matrix and systems of linear equations

(c) A company manufactures two products: X and Y. Product X costs $8 per unit and contains 2 units of raw material 1, 6 units of raw material 2 and 4 units of raw material 3. Product Y costs $10 per unit and contains 2 units of raw material 1, 2 units of raw material 2 and 12 units of raw material 3. The company has 80 units of raw material 1, 120 units of raw material 2 and 240 units of raw material 3 available. In addition, the product Y should be at least 15 units less than product X. How many units of each product should the company produce to minimize the cost of production? (i) Derive the system of equations. (4 marks) (ii) Use the extreme point method to solve the system. Ensure that you explain each step in the computation. (7 marks)(a) Consider an optimization problem (i) How are the augmented matrices for a minimization and dual maximization problem related? Explain. (3 marks) (ii) What is the impact of changing the right-hand side coefficients on the optimal solution? Explain