1 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 4-1 Systems of Equations and Inequalities Chapter 4

2 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter – Solving Systems of Linear Equations in Two Variables 4.2 – Solving Systems of Linear Equations in Three Variables 4.3 – Systems of Linear Equations: Applications and Problem Solving 4.4 – Solving Systems of Equations Using Matrices 4.5 – Solving Systems of Equations Using Determinants and Cramer’s Rule 4.6 – Solving Systems of Linear Inequalities Chapter Sections

3 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 4-3 § 4.2 Solving Systems of Linear Equations in Three Variables

4 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 4-4 Definitions The equation 2x – 3y + 4z = 8 is an example of a linear equation in three variables. The solution to this type of equation is an ordered triple of the form (x, y, z). One possible solution to the equation 5x – 3y + 4z = 9 is (1, 2, 3).

5 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 4-5 Solving Systems Systems in three (or more) linear equations are solved the same way systems of two linear equations are solved by using either the substitution or addition method. Solve the following system of equations using the substitution method.

6 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 4-6 Solving Systems Since we know that x = -3, we can substitute it into the equation 3x + 4y = 7 and solve for y.

7 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 4-7 Solving Systems Now we substitute x=-3 and y=4 into the last equation and solve for z. The solution is the ordered triple (-3, 4, 5).

8 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 4-8 x y z (4, 5, 3) Geometric Interpretation The following is a geometric interpretation of the solution (4, 5, 3).

9 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 4-9 Inconsistent and Dependent Systems Inconsistent System of Equations A system that has no solution. Example: You obtain a statement that is always false, such as 3=0. Dependent System of Equations A system that has an infinite number of solutions. Example: You obtain a statement that is always true, such as 0=0.