Financial Literacy is the capability and understanding of money and can manage it.
The best and most correct answer among the choices provided by your question is the fourth choice or letter D.
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Answer:
(a) Linear model
![max\ P = 9x + 7y](https://tex.z-dn.net/?f=max%5C%20P%20%3D%209x%20%2B%207y)
Subject to:
![12x + 4y \le 60](https://tex.z-dn.net/?f=12x%20%2B%204y%20%5Cle%2060)
![4x + 8y \le 40](https://tex.z-dn.net/?f=4x%20%2B%208y%20%5Cle%2040)
![x,y \ge 0](https://tex.z-dn.net/?f=x%2Cy%20%5Cge%200)
(b) Standard form:
![max\ P = 9x + 7y](https://tex.z-dn.net/?f=max%5C%20P%20%3D%209x%20%2B%207y)
Subject to:
![12x + 4y + s_1 = 60](https://tex.z-dn.net/?f=12x%20%2B%204y%20%2B%20s_1%20%3D%2060)
![4x + 8y +s_2= 40](https://tex.z-dn.net/?f=4x%20%2B%208y%20%2Bs_2%3D%2040)
![x,y \ge 0](https://tex.z-dn.net/?f=x%2Cy%20%5Cge%200)
![s_1,s_2 \ge 0](https://tex.z-dn.net/?f=s_1%2Cs_2%20%5Cge%200)
Explanation:
Given
![\begin{array}{ccc}{} & {Hours/} & {Unit} & {Product} & {Line\ 1} & {Line\ 2} & {A} & {12} & {4} & {B} & {4} & {8} & {Total\ Hours} & {60} &{40}\ \end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bccc%7D%7B%7D%20%26%20%7BHours%2F%7D%20%26%20%7BUnit%7D%20%26%20%7BProduct%7D%20%26%20%7BLine%5C%201%7D%20%26%20%7BLine%5C%202%7D%20%26%20%7BA%7D%20%26%20%7B12%7D%20%26%20%7B4%7D%20%26%20%7BB%7D%20%26%20%7B4%7D%20%26%20%7B8%7D%20%26%20%7BTotal%5C%20Hours%7D%20%26%20%7B60%7D%20%26%7B40%7D%5C%20%5Cend%7Barray%7D)
Solving (a): Formulate a linear programming model
From the question, we understand that:
A has a profit of $9 while B has $7
So, the linear model is:
![max\ P = 9x + 7y](https://tex.z-dn.net/?f=max%5C%20P%20%3D%209x%20%2B%207y)
Subject to:
![12x + 4y \le 60](https://tex.z-dn.net/?f=12x%20%2B%204y%20%5Cle%2060)
![4x + 8y \le 40](https://tex.z-dn.net/?f=4x%20%2B%208y%20%5Cle%2040)
![x,y \ge 0](https://tex.z-dn.net/?f=x%2Cy%20%5Cge%200)
Where:
![x \to line\ 1](https://tex.z-dn.net/?f=x%20%5Cto%20line%5C%201)
![y \to line\ 2](https://tex.z-dn.net/?f=y%20%5Cto%20line%5C%202)
Solving (b): The model in standard form:
To do this, we introduce surplus and slack variable "s"
For
inequalities, we add surplus (add s)
Otherwise, we remove slack (minus s)
So, the standard form is:
So, the linear model is:
![max\ P = 9x + 7y](https://tex.z-dn.net/?f=max%5C%20P%20%3D%209x%20%2B%207y)
Subject to:
![12x + 4y + s_1 = 60](https://tex.z-dn.net/?f=12x%20%2B%204y%20%2B%20s_1%20%3D%2060)
![4x + 8y +s_2= 40](https://tex.z-dn.net/?f=4x%20%2B%208y%20%2Bs_2%3D%2040)
![x,y \ge 0](https://tex.z-dn.net/?f=x%2Cy%20%5Cge%200)
![s_1,s_2 \ge 0](https://tex.z-dn.net/?f=s_1%2Cs_2%20%5Cge%200)