The minimum value of 3𝑥 + 5𝑦 such that:

3𝑥 + 5𝑦 ≤ 15

4𝑥 + 9𝑦 ≤ 8

13𝑥 + 2𝑦 ≤ 2

𝑥 ≥ 0, 𝑦 ≥ 0  is _______________

To find the minimum value of the objective function, let us denote it by $Z$:

$Z=3x+5y$

We are given the following constraints:

$3x+5y\le 15$

$4x+9y\le 8$

$13x+2y\le 2$

$x\ge 0,y\ge 0$

Let us analyze the system step-by-step to find the minimum value:

Step 1: Check the non-negativity constraints
The variables $x$ and $y$ must be non-negative: $x\ge 0$ and $y\ge 0$. This restricts our search for the feasible region to the first quadrant of the coordinate plane.

Step 2: Understand the behavior of the objective function
The objective function is $Z=3x+5y$. Since the coefficients of both variables (3 and 5) are strictly positive, the value of $Z$ increases as $x$ or $y$ increases. Conversely, $Z$ decreases as $x$ and $y$ approach their minimum possible values.

Since $x\ge 0$ and $y\ge 0$, the minimum possible value that the term $3x+5y$ can mathematically achieve is when both variables are zero:
$x=0$ and $y=0$

Step 3: Verify if the origin is in the feasible region
We must check if the point $(0,0)$ satisfies all the given inequality constraints:
1. For $3x+5y\le 15$:
$3\left(0\right)+5\left(0\right)=0\le 15$ (Satisfied)
2. For $4x+9y\le 8$:
$4\left(0\right)+9\left(0\right)=0\le 8$ (Satisfied)
3. For $13x+2y\le 2$:
$13\left(0\right)+2\left(0\right)=0\le 2$ (Satisfied)
4. For $x\ge 0,y\ge 0$:
$0\ge 0,0\ge 0$ (Satisfied)

Since the point $(0,0)$ satisfies all the constraints, it lies within the feasible region.

Step 4: Calculate the minimum value of Z
Evaluating the objective function at the origin:
$Z=3\left(0\right)+5\left(0\right)=0$

Since $Z\ge 0$ for all points in the feasible region, the absolute minimum value of the objective function is $0$ at the point $(0,0)$.