In Problems 9–18, solve each linear programming problem. Maximize z=5 x+3 y subject to x ≥0,

In Problems 9–18, solve each linear programming problem. ...

Key Concepts

Linear Programming

Linear programming is an optimization technique used to maximize or minimize a linear objective function, subject to a set of linear inequality or equality constraints. It plays a crucial role in operational research and various decision-making processes where resources need to be allocated efficiently.

Objective Function

The objective function in linear programming is a mathematical expression that needs to be maximized or minimized. It represents the goal of the optimization problem, such as profit maximization or cost minimization, and it is defined as a linear combination of decision variables.

Constraints

Constraints are the restrictions or limitations imposed on the decision variables in a linear programming problem. These conditions, expressed as linear inequalities or equalities, restrict the feasible values of the variables and define the boundaries of the solution space.

Feasible Region

The feasible region is the set of all possible points that satisfy all of the problem's constraints. In graphical linear programming, this region is often depicted as the intersection of half-planes, and the optimal solution is typically found at one of its vertices.

Graphical Method

The graphical method is a technique used to solve linear programming problems involving two variables by plotting the constraints on a coordinate system. It helps visualize the feasible region and identifies the corner points where the optimal solution can occur.

Corner-Point Rule

The corner-point rule, or the vertex theorem, states that the optimal solution to a linear programming problem, if it exists, will be found at one of the vertices (or corner points) of the feasible region. This principle simplifies the process of finding the optimum by evaluating the objective function at these critical points.

Transcript

00:01 So you want to solve this linear programming problem, and we're already given our objective function as well as our constraints.

00:08 We just need to graph them.

00:10 So these two inequalities tell us that our graph will lie completely in our first quadrant.

00:17 Now, this inequality here i can rewrite as y is greater than or equal to negative x plus 2, which i'll graph with a y intercept of 2 and x intercept of 2 and a slope of negative 1.

00:31 This here, similarly, i can graph as y is less than or equal to negative x plus 8.

00:38 So our y intercept will be 8.

00:41 Our x intercept will also be 8 because our slope is negative 1.

00:46 And our line will look like so.

00:50 And now we have this function, which we can rewrite as y is less than or equal to negative 2x plus 10, which will graph with a...

01:01 Y intercept of 10 and since our slope is negative 2 we have our x intercept at x equals 5 and it should look like this and we can shade in the area which tells us that this area is where our feasible points lie and now we want to find the coordinates of each of the corners of this graph and we can start with this corner right here, which is 0, 2.

01:39 We move up here, this corner is 0 .8.

01:45 And here we're not entirely sure, so what we can do is we can set the equations of these two lines, which would be y is lessen or equal to negative 2x plus 10, and y is less than or equal to negative x plus 8.

02:00 We can set those equal to each other.

02:01 So we solve for negative x plus 8 equals negative 2x plus 10.

02:09 So i'm going to add 2x to both sides to get x plus 8 equals 10, x equals 2...

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