Solving a Linear Programming Problem, use a graphing utility...

Solving a Linear Programming Problem, use a graphing utility to graph the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the constraints. $$ \begin{array}{c}{\text { Objective function: }} \\ {z=3 x+y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {x+4 y \leq 60} \\ {3 x+2 y \geq 48}\end{array} $$

Solving a Linear Programming Problem, use a graphing utility to graph the
region determined by the constraints. Then find the
minimum and maximum values of the objective function
and where they occur, subject to the constraints.
$$
\begin{array}{c}{\text { Objective function: }} \\ {z=3 x+y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {x+4 y \leq 60} \\ {3 x+2 y \geq 48}\end{array}
$$

Key Concepts

Linear Programming

Linear programming is a mathematical optimization technique that focuses on maximizing or minimizing a linear objective function subject to a set of linear constraints. It is widely used to solve problems where the resources are limited and must be allocated in the most efficient way.

Feasible Region

The feasible region is the collection of all points that satisfy the set of constraints in a linear programming problem. It represents all potential solutions and is typically depicted graphically as a polygonal area in the case of two-variable problems.

Graphical Method

The graphical method is a technique used to solve linear programming problems visually by plotting the constraint lines on a coordinate plane. The intersection and shading of these lines help identify the feasible region, making it easier to analyze and solve the problem.

Objective Function Optimization

The objective function in linear programming is the function to be maximized or minimized, representing the goal of the optimization problem. The optimal solution is determined by evaluating this function at key candidate points within the feasible region.

Corner-Point Theorem

The corner-point theorem, a fundamental principle in linear programming, states that if an optimal solution exists, it will occur at one of the vertices (corner points) of the feasible region. This theorem underpins the strategy of evaluating the objective function at these points to find the maximum or minimum value.

Transcript

00:01 In this problem we are trying to solve the linear programming problem.

00:05 So this is y xx and this is xx.

00:10 We need to graph our constraints first.

00:13 So this is x is greater than equal to 0 and this is y is greater than equal to 0.

00:22 Then next we have x plus 4 y is equal to 60.

00:27 We need x and y intercepts to graph our equation so the x intercept at x intercept y is 0 so we have x plus 0 is equal to 60 so x intercept is 60 60 comma 0 then at y intercept x is 0 so 0 plus 4 y is equal to 60 so y is equal to 15 and r y intercept is 0 and if we join these two points we get our graph and we have x plus 4 y less than equal to 60 so less than means shade is downwards this is our shade then we have 3x plus 2 y is greater than equal to 48.

01:35 So again we need x and y intercept so 3x is equal to 48 gives x is equal to 16.

01:44 So the x intercept is 16.

01:48 0 and 2 y is equal to 48 gives y is equal to 24 and y intercept is 0 4 .24.

02:01 Now we join these two points and this is our graph but we need shade as well and we have 3x plus 2y is greater than equal to 48 greater than means we have shade upwards so the common region is this region.

02:28 The triangle this is the common region in the shape.

02:34 We also need the point of intersection.

02:38 So the equations we have are x plus 4y is equal to 60 and 3x plus 2y is equal to 48...

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