Find the vertices of a solution set

Learning Objectives Accurately graph a system of linear inequalities on the Cartesian plane. Identify the feasible region (solution set) for a given system of inequalities. Algebraically determine the coordinates of each vertex of the feasible region. Solve the system of linear equations corresponding to the boundary lines that form a vertex. Distinguish between bounded and unbounded solution sets. Verify that a calculated vertex satisfies all inequalities in the system. How does a business determine the perfect production level to maximize profit while minimizing waste? 🤔 It all starts by finding the corners of their operational possibilities! This tutorial will guide you through the process of finding the vertices, or corner points, of a solution set defined by a system...

TermDefinitionExample System of Linear InequalitiesA collection of two or more linear inequalities involving the same variables. The solution must satisfy all inequalities simultaneously.y ≤ 2x + 1, y > -x + 3, x ≥ 0 Feasible Region (Solution Set)The area on the coordinate plane containing all the points (x, y) that are solutions to every inequality in the system. It is the region where all shaded areas overlap.For the system x ≥ 0, y ≥ 0, and x + y ≤ 4, the feasible region is a triangle with corners at (0,0), (4,0), and (0,4). Boundary LineThe line that corresponds to the equation part of an inequality. It is drawn as a solid line for ≤ or ≥, and a dashed line for < or >.For the inequality y ≤ 3x - 2, the boundary line is the equation y = 3x - 2. Vertex (Corner Point)A point whe...

Finding a Vertex To find a vertex, identify the two boundary lines that intersect at that point. Set their corresponding equations equal to each other and solve the resulting system of two linear equations. This is the core algebraic method for finding the exact coordinates of a vertex. For a system with equations y = m₁x + b₁ and y = m₂x + b₂, you would solve m₁x + b₁ = m₂x + b₂. Test Point Method 1. Graph the boundary line. \n2. Choose a test point not on the line (e.g., (0,0)). \n3. Substitute the point into the original inequality. \n4. If the inequality is true, shade the region containing the test point. If false, shade the other side. Use this method to determine which side of a boundary line represents the solution for that inequality. This ensures you correctly iden...