Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 1 Chapter 3 Systems of Linear Equations

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide Using Graphs and Tables to Solve Systems

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 3 Example: Using Two Models to Make a Prediction In the United States, life expectancies of women have been longer than life expectancies of men for many years. Data is shown in the table on the next slide. The life expectancies (in years) W(t) and M(t) of women and men, respectively, are modeled by the system L = W(t) = 0.115t L = M(t) = 0.208t = where t is the number of years since 1980.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 4 Example: Using Two Models to Make a Prediction Use graphs of W and M to predict when life expectancies of women and men will be equal.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 5 Solution Begin by sketching graphs of W and M on the same coordinate system.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 6 Solution The intersection point is approximately (81.51, 86.81). So, the models predict that the life expectancy of both women and men will be about 86.8 years in We are not very confident about this prediction, however, because it is so far into the future.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 7 Solution We verify our work using “intersect” on a graphing calculator.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 8 Intersection Point of the Graphs of Two Models If the independent variable of two models represents time, then an intersection point of the graphs of the two models indicates a time when the quantities represented by the dependent variables were or will be equal.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 9 Systems of Two Linear Equations A system of linear equations in two variables, or a linear system for short, consists of two or more linear equations in two variables.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 10 Solution of a system Definition We say an ordered pair (a, b) is a solution of a system of two equations in two variables if it satisfies both equations. The solution set of a system is the set of all solutions of the system. We solve a system by finding its solution set.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 11 Solution Set In general, the solution set of a system of two linear equations can be found by locating any intersection point(s) of the graphs of the two equations.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 12 Example: Solving a System of Two Linear Equations by Graphing Solve the system y = 2x + 4 y = –x + 1

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 13 Solution The graphs of the equations are shown at the right. The intersection point is (–1, 2). So, the solution is the ordered pair (–1, 2).

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 14 Solution We can verify that (–1, 2) satisfies both equations:

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 15 Solution We can also verify our work using “intersect” on a graphing calculator.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 16 Solving a System Warning It is important to check that your result satisfies both equations.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 17 Example: Solving an Inconsistent System Solve the system

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 18 Solution Since the two lines have equal slopes, these lines are parallel. Parallel lines do not intersect, so there is no ordered pair that satisfies both equations. The solution set is the empty set.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 19 Inconsistent System A linear system whose solution set is the empty set is called an inconsistent system.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 20 Example: Solving a Dependent System Solve the system

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 21 Solution Write the second equation in slope-intercept form:

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 22 Solution So, the graphs of 6x – 3y = –3 and y = 2x + 1 are the same line. The solution set of the system is the set of the infinite number of ordered pairs that correspond to points that lie on the line y = 2x + 1 and on the (same) line 6x – 3y = –3.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 23 Dependent System A linear system that has an infinite number of solutions is called a dependent system.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 24 Types of Linear Systems There are three types of linear systems of two equations: 1. One-solution system: The lines intersect in one point. The solution set of the system contains only the ordered pair that corresponds to that point.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 25 Types of Linear Systems 2. Inconsistent system: The lines are parallel. The solution set of the system is the empty set.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 26 Types of Linear Systems 3. Dependent system: The lines are identical. The solution set of the system is the set of the infinite number of solutions represented by all points on the same line.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 27 Example: Using a Table to Solve a System Use a table of solutions to solve the following system of two equations: y = 2x – 3 y = –3x + 7

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 28 Solution Some solutions of the two equations are shown in the table below. Since the ordered pair (2, 1) is a solution of both equations, it is a solution of the system of equations. The lines have different slopes, so there is only one intersection point. Thus, the ordered pair (2, 1) is the only solution of the system.

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 3.1, Slide 29 Using a Table to Solve a System If an ordered pair is listed in a table as a solution of both of two linear equations, then that ordered pair is a solution of the system.