Maximize C= 4x + 12y, Subject to 3x + 5y <= 12, 6x + 2y <= 10, x >= 0, y >= 0

We are tasked with maximizing the objective function:

\[
C = 4x + 12y
\]

subject to the constraints:

\[
3x + 5y \leq 12
\]
\[
6x + 2y \leq 10
\]
\[
x \geq 0, \quad y \geq 0
\]

### Step 1: Identify the constraints

We need to convert the inequalities into equations for the boundary lines and then identify the feasible region formed by these constraints.

1. \( 3x + 5y = 12 \) — Constraint 1
2. \( 6x + 2y = 10 \) — Constraint 2
3. \( x \geq 0, y \geq 0 \) — Non-negativity constraints.

### Step 2: Find the intersection points

Let's find the points of intersection between the constraints by solving the system of equations:

#### Intersection of \( 3x + 5y = 12 \) and \( 6x + 2y = 10 \):

Multiply the second equation by 2 to eliminate \(x\):

\[
12x + 4y = 20
\]

Now subtract the first equation:

\[
(12x + 4y) - (9x + 15y) = 20 - 12
\]
\[
3x - 11y = 8
\]

Solve for \(x\):

\[
x = \frac{8 + 11y}{3}
\]

Substitute into \(6x + 2y = 10\) or simplify intersection.

Maximize the objective function C= 4x + 12y subject to linear inequalities 3x + 5y <= 12, 6x + 2y <= 10, and non-negativity constraints x >= 0, y >= 0. This linear programming problem involves finding intersection points of boundary lines and optimizing within the feasible region. Suitable for advanced high school or college-level students.

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