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 1

Chapter 3Systems of Linear Equations


3.1 Using Graphs and Tables to Solve Systems


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 + 77.44L = M(t) = 0.208t = 69.86

where t is the number of years since 1980.


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.


Solution

Begin by sketching graphs of W and M on the same coordinate system.


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 2062.

We are not very confident about this prediction, however, because it is so far into the future.


Solution

We verify our work using “intersect” on a graphing calculator.


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.


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.


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.


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.


Example: Solving a System of Two Linear Equations by Graphing

Solve the system

y = 2x + 4y = –x + 1


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).


Solution

We can verify that (–1, 2) satisfies both equations:


Solution

We can also verify our work using “intersect” on a graphing calculator.


Solving a System

Warning

It is important to check that your result satisfies both equations.


Example: Solving an Inconsistent System

Solve the system

11

2y x

12

2y x


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.


Inconsistent System

A linear system whose solution set is the empty set is called an inconsistent system.


Example: Solving a Dependent System

Solve the system

2 1y x

6 3 3x y


Solution

Write the second equation in slope-intercept form:

6 3 3x y

2 1x y

2 1y x

2 1y x


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.


Dependent System

A linear system that has an infinite number of solutions is called a dependent system.


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.



2. Inconsistent system: The lines are parallel. The solution set of the system is the empty set.



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.


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


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.


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.

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