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# Chapter 5

### Systems

What is the Addition Property of Inequality Multiplication Properties of Inequality? For all real numbers a, b, and c: if a is less than b, than a+c is less than b+c. If a is less than b and c is greater than 0 then ac is less than bc. If a is less than b and c is less than 0, then ac is greater than bc.
What is a consistent system? A system which has one or more solutions is called a consistent system.
What is an inconsistent system? A system which has no solutions is called an inconsistent system.
What is the intersection of two sets? The intersection of two sets is the set consisting of those values common to both sets.
What is the union of two sets? The union of two sets is the set consisting of those values in either one or both sets.
What is the solution set for a system? The solution set for a system is the intersection of the solution sets for the individual sentences. When the system involves two equations each with two variables, the solution can often be found by making a table of values or a graph.
What is a system-determinant theorem? A 2 X 2 system has exactly one solution if and only if the determinant of the coefficient matrix is not zero.
What is a half-planes? When a line is drawn in a plane, the line separates the plane into two distinct regions is called half-planes. The line itself is the boundary of the two regions. The boundary does not belong to either half plane.
What is a feasible set or feasible region? The set of solutions to a system of linear inequalities is often called feasible set or feasible region for that system.
What is the linear-programming theorem? The feasible region of every linear-programming problem is convex, and the maximum or minimum quantity is determined at one of the vertices of this feasible region.
What steps should be followed to solve a linear-programming problem? Follow these steps: 1. Identify the variables. 2. Identify the contraints, and translate them into a system of inequalities relating the variables. 3. Graph the system. 4.Write a formula or an expression.5.Apply the theorem. 6. Interpret the results.
Created by: Southlawn