In Problems 9–18, solve each linear programming problem. Minimize z=2 x+3 y subject to x ≥0,

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

Key Concepts

Linear Programming

Linear programming is a method used to optimize a linear objective function subject to a set of linear constraints. This approach is widely used in various fields to determine the best outcome, such as minimum cost or maximum profit, within a given set of limitations.

Objective Function

The objective function is a linear expression that needs to be optimized, either minimized or maximized, in a linear programming problem. It represents the quantity that one aims to optimize, such as cost, time, or profit.

Constraints

Constraints are the linear inequalities or equations that define the restrictions under which the objective function must be optimized. They limit the values that the decision variables can take, forming the boundaries of the feasible region where a solution exists.

Feasible Region

The feasible region is the set of all points that satisfy a linear programming problem's constraints. It is typically formed by the intersection of half-planes defined by these constraints, and it represents all possible solutions that meet the problem's requirements.

Corner Point Theorem

The Corner Point Theorem states that if an optimal solution exists for a linear programming problem, then at least one of the optimal solutions will be found at a vertex, or corner point, of the feasible region. This principle is vital for solving linear programming problems, especially when using graphical methods.

Transcript

00:01 We're given our objective function z equals 2x plus y as well as all of our constraints, and we want to begin solving this linear programming problem by graphing our constraints.

00:12 So we're given that x and y are both greater than or equal to zero, meaning that our graph will lie solely in the first quadrant.

00:21 And now i want to start graphing these lines.

00:23 So x plus y greater than or equal to 3, i can rewrite this as y is greater than or equal to negative x plus 3.

00:31 This makes it easier for me to graph, and i know that my y intercept is 3.

00:36 I have a slope of negative 1, and my x intercept would also be 3.

00:42 Similarly here, we know we can rewrite this as y is less than or equal to negative x plus 9.

00:50 So my y intercept is at 9.

00:54 I have a slope again of negative 1, meaning that our x intercept would also be at 9.

01:01 Here we have 3y is greater than or equal to negative x plus 6.

01:08 Well, we can divide both sides by 3 to isolate the y, and we get y is greater than or equal to negative 1 over 3x plus 2.

01:18 So we know that our y intercept is at 2.

01:21 We have a slope of negative 1 3rd, which means that our x intercept would be at 6, and our line looks like this.

01:32 Now the space in between these lines is our collection of feasible points.

01:41 And given these, we want to find the coordinates of all of our corner points and write them down.

01:47 So starting at this corner, i know that that point is 0 .3.

01:52 Moving here, this corner is 0 .9.

01:56 Over here, that corner is 9 .0.

02:01 This corner here is 6 .0.

02:06 And to solve for the coordinates of this corner, we're going to set the equations y is equal to negative 1 3x plus 2 as well as y is equal to negative x plus 3 equal to each other.

02:23 So negative x plus 3 equals negative 1 over 3x plus 2...

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