Question Given constraints and optimal solution $(x, y) = (25, 0)$ , find the values of the slack variables $s_1$ and $s_2$ .

Studdy Solution

STEP 1

Assumptions
1. The first constraint is $2x + 4y \leq 1440$ .
2. The second constraint is $46x + 35y \leq 1150$ .
3. The non-negativity constraints are $x \geq 0$ and $y \geq 0$ .
4. The optimal solution given is $(x, y) = (25, 0)$ .
5. Slack variables $s_1$ and $s_2$ represent the unused resources in the first and second constraints, respectively.

STEP 2

To find the value of the slack variable $s_1$ associated with the first constraint, we substitute the optimal solution $(x, y) = (25, 0)$ into the first constraint equation.
$s_1 = 1440 - (2x + 4y)$

STEP 3

Now, plug in the values for $x$ and $y$ into the equation for $s_1$ .
$s_1 = 1440 - (2 \cdot 25 + 4 \cdot 0)$

STEP 4

Calculate the value of $s_1$ .
$s_1 = 1440 - (50 + 0)$ $s_1 = 1440 - 50$ $s_1 = 1390$

STEP 5

To find the value of the slack variable $s_2$ associated with the second constraint, we substitute the optimal solution $(x, y) = (25, 0)$ into the second constraint equation.
$s_2 = 1150 - (46x + 35y)$

STEP 6

Now, plug in the values for $x$ and $y$ into the equation for $s_2$ .
$s_2 = 1150 - (46 \cdot 25 + 35 \cdot 0)$

STEP 7

Calculate the value of $s_2$ .
$s_2 = 1150 - (1150 + 0)$ $s_2 = 1150 - 1150$ $s_2 = 0$

STEP 8

The values of the slack variables are $s_1 = 1390$ and $s_2 = 0$ .
The slack variables have values $s_{1}=1390$ and $s_{2}=0$ .

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