In Problems 9–18, solve each linear programming problem. Maximize z=5 x+2 y subject to x ≥0,

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

In Problems 9–18, solve each linear programming problem.
$$
\text { Maximize } z=5 x+2 y \text { subject to } x \geq 0, \quad y \geq 0, \quad x+y \leq 10, \quad 2 x+y \geq 10, \quad x+2 y \geq 10
$$

Key Concepts

Linear Programming

Linear programming is an optimization technique used to achieve the best outcome, such as maximum profit or lowest cost, in a mathematical model whose requirements are represented by linear relationships. This subject area uses methods to maximize or minimize a linear objective function subject to a set of linear constraints.

Objective Function

The objective function in linear programming is the linear equation that needs to be maximized or minimized. It represents the measure of performance or cost, and its value is determined by the decision variables in the problem.

Constraints

Constraints in linear programming are the linear equations or inequalities that restrict the values of the decision variables. They define the conditions that any solution must satisfy, setting the boundaries within which the optimal solution must be found.

Feasible Region

The feasible region is the set of all possible points that satisfy the constraints of a linear programming problem. It represents all viable solutions, and the optimal solution is typically found at one of the vertices of this region.

Graphical Method

The graphical method is a technique used to solve linear programming problems with two decision variables. By plotting the constraints on a graph, the feasible region is identified, and the optimal value of the objective function is determined by evaluating it at the vertices of this region.

Corner Point Theorem

The corner point theorem states that if an optimal solution to a linear programming problem exists, it will be found at one of the vertices, or corner points, of the feasible region. This theorem is fundamental in simplifying the search for the optimal solution.

Transcript

00:01 So in this linear programming problem, we're given our objective function as well as our constraints.

00:07 And we want to start grapping the constraints.

00:10 To start off, we know that x and y are both greater than or equal to zero, meaning that our graph lies solely in the first quadrant.

00:18 Now we want to start grapping these lines down here.

00:21 So we can rewrite this as y is less than or equal to negative x plus 10, just to make it easier for us to graph.

00:30 So our y intercept is at 10.

00:33 Our slope is negative 1, meaning that our x intercept will also be at 10.

00:40 Here we have y is greater than or equal to negative 2x plus 10.

00:47 And our y intercept is also at 10, but our x intercept would be at 5 because our slope is negative 2.

00:54 Underline would look something like this.

00:58 And now we rearrange this equation to be 2.

01:02 2y is greater than or equal to negative x plus 10.

01:06 But we can divide both sides by 2 to get y is greater than or equal to negative 1 half x plus 5.

01:15 So our y intercept would be at 5 this time.

01:19 And since we have a slope of negative 1 1 half, we know that our x intercept would be at 10.

01:25 So our feasible points would lie in this area between the the lines.

01:35 And now what we do is we want to find the coordinates of all of our corner points and list them...

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