Copyright © 2011 Pearson Education, Inc. Systems of Linear Equations in Two Variables Section 5.1 Systems of Equations and Inequalities

5.1 Copyright © 2011 Pearson Education, Inc. Slide 5-3 A linear equation in two variables is an equation of the form Ax + By = C, where A and B are not both zero. There are infinitely many ordered pairs that satisfy a single linear equation. We are often interested in finding a single ordered pair that satisfies a pair of linear equations. Any collection of two or more equations is called a system of equations. The solution set of a system of two linear equations in two variables is the set of all ordered pairs that satisfy both equations of the system. The graph of an equation shows all ordered pairs that satisfy the equation, so we can solve some systems by graphing the equations and observing which points (if any) satisfy all of the equations. Solving a System by Graphing

5.1 Copyright © 2011 Pearson Education, Inc. Slide 5-4 A system of equations that has at least one solution is consistent. There are two types of consistent systems. A consistent system with exactly one solution is independent. A consistent system with infinitely many solutions is dependent. A system with no solutions is inconsistent. Types of Systems

5.1 Copyright © 2011 Pearson Education, Inc. Slide 5-5 In the substitution method, we eliminate a variable from one equation by substituting an expression for that variable from the other equation. In the addition method, we eliminate a variable by adding the two equations. It might be necessary to multiply each equation by an appropriate number so that a variable will be eliminated by this addition. The Substitution Method and the Addition Method