5.3 Q-5

Question Statement

Maximize the function $f(x, y) = 2x + 3y$ subject to the following constraints:

$2x + y \leq 8$
$x + 2y \leq 14$
$x \geq 0$
$y \geq 0$

Background and Explanation

To solve this problem, we will use Linear Programming. The objective is to find the values of $x$ and $y$ that maximize the function $f(x, y)$ while satisfying all the given constraints. The feasible region is formed by the intersection of the graphs of the constraints, and the optimal solution occurs at one of the corner points of this region.

Solution

Start with the first constraint:

2x + y \leq 8 \quad \text{(Equation 1)}

Rewrite it as an equality to find the boundary:

2x + y = 8

Now, substitute $x = 0$ to find $y$:

0 + y = 8 \quad \Rightarrow \quad y = 8 \quad \text{(Point (0, 8))}

Next, substitute $y = 0$ to find $x$:

2x + 0 = 8 \quad \Rightarrow \quad x = 4 \quad \text{(Point (4, 0))}

The inequality $0 + 0 < 8$ is true, confirming the feasibility of the origin.

2. Now, work with the second constraint:

x + 2y \leq 14 \quad \text{(Equation 2)}

Rewrite this as an equality:

x + 2y = 14

Substitute $x = 0$ to find $y$:

0 + 2y = 14 \quad \Rightarrow \quad y = 7 \quad \text{(Point (0, 7))}

Substitute $y = 0$ to find $x$:

x + 0 = 14 \quad \Rightarrow \quad x = 14 \quad \text{(Point (14, 0))}

The inequality $0 + 0 < 14$ is true, confirming the feasibility of the origin.

3. Find the intersection of the two lines (Equation 1 and Equation 2): - From Equation 2, we express $x$ as:

x = 14 - 2y \quad \text{(Equation 3)}

- Substitute this into Equation 1:

2(14 - 2y) + y = 8

Simplify the equation:

28 - 4y + y = 8 \quad \Rightarrow \quad -3y = -20 \quad \Rightarrow \quad y = \frac{20}{3}

- Now substitute $y = \frac{20}{3}$ back into Equation 3 to find $x$:

x = 14 - 2\left(\frac{20}{3}\right) = 14 - \frac{40}{3} = \frac{42}{3} - \frac{40}{3} = \frac{2}{3}

Therefore, the intersection point is $\left(\frac{2}{3}, \frac{20}{3}\right)$.

4. Identify the corner points: The feasible region is bounded by the points $(0, 8)$ , $(4, 0)$ , $(0, 7)$ , and $\left(\frac{2}{3}, \frac{20}{3}\right)$ .

Key Formulas or Methods Used

Linear Programming: Finding the optimal solution by analyzing the feasible region created by constraints and evaluating the objective function at the corner points.
Substitution: Used to find the intersection points of the constraint lines.

Summary of Steps

Write the constraints as equalities to find the boundary lines.
Find the intersection points of the lines by solving the system of equations.
Identify the feasible region and its corner points.
Evaluate the objective function $f(x, y) = 2x + 3y$ at each corner point.
Select the corner point that maximizes the objective function.

Maximum Value: The maximum value of the function occurs at the corner point $\left(\frac{2}{3}, \frac{20}{3}\right)$ , where the function value is $\frac{64}{3} \approx 21.33$ .