Copyright © 1999 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen Precalculus: Functions & Graphs, 4 th Edition Chapter Eight Systems of Equations & Inequalities

x y 5–5 5 (4, 2) x y 5–5 5 x y 5 – 5 5 (A) 2x – 3y= 2 x + 2y= 8 (B) 4x + 6y= 12 2x + 3y= –6 Lines intersect at one point only. Exactly one solution: x = 4, y = 2 Lines are parallel (each has slope –  ). No solution. (C) 2x – 3y = –6 –x + 3 / 2 y = 3 Lines coincide. Infinitely many solutions. Nature of Solutions to Systems of Equations

In general, associated with each linear system of the form a 11 x 1 + a 12 x 2 = k 1 a 2 1 x 1 + a 22 x 2 = k 2 where x 1 and x 2 are variables, is the augmented matrix of the system: Augmented Matrix

An augmented matrix is transformed into a row-equivalent matrix if any of the following row operations is performed: 1. Two rows are interchanged (R i  R j ). 2. A row is multiplied by a nonzero constant (kR i  R i ). 3. A constant multiple of another row is added to a given row (kR j + R i  R i ). [Note: The arrow means "replaces."] Elementary Row Operations Producing Row-Equivalent Matrices

Reduced Matrix

Gauss-Jordan Elimination

x y 1 –1 1 x y 5–5 5 x y x 2 + y 2 = 5 3x + y= 1 Two real solutions. 2. x 2 – 2y 2 = 2 xy = 2 Two real solutions and two imaginary solutions. (Imaginary solutions cannot be shown on the graph.) 3. x 2 + 3xy + y 2 = 20 xy – y 2 = 0 Four real solutions. Solutions of Nonlinear Systems of Equations

(a) y  2x – 3 (b) y > 2x – 3 (c) y  2x – 3 (d) y < 2x – 3 Graph of a Linear Inequality

Procedure for Graphing Linear Inequalities

Solution of Linear Programming Problems