Download solve a system of equations

Section 4.1
Solving Systems of
Equations by Graphing
Objectives
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



Determine whether a given ordered pair is a solution
of a system
Solve systems of linear equations by graphing
Use graphing to identify inconsistent systems and
dependent equations
Identify the number of solutions of a linear system
without graphing
Use a graphing calculator to solve a linear system
(optional)
Objective 1: Determine Whether a Given
Ordered Pair Is a Solution of a System
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We have previously discussed equations in two
variables, such as x + y = 3. Because there are
infinitely many pairs of numbers whose sum is 3,
there are infinitely many pairs (x, y) that satisfy
this equation. Some of these pairs are listed in
table (a).
Now consider the equation x − y = 1. Because
there are infinitely many pairs of numbers whose
difference is 1, there are infinitely many pairs
(x, y) that satisfy x − y = 1. Some of these pairs
are listed in table (b).
From the two tables, we see that (2, 1) satisfies
both equations.
Objective 1: Determine Whether a Given
Ordered Pair Is a Solution of a System

When two equations with the same variables are
considered simultaneously (at the same time), we say
that they form a system of equations.
Using a left brace { , we can write the equations from the
previous example as a system:

Because the ordered pair (2, 1) satisfies both of these
equations, it is called a solution of the system.

In general, a system of linear equations can have exactly
one solution, no solution, or infinitely many solutions.
EXAMPLE 1

Determine whether (−2, 5) is a
solution of each system of equations.
Objective 2: Solve Systems of Linear
Equations by Graphing

To solve a system of equations means to find all
of the solutions of the system.
 One
way to solve a system of linear equations is to
graph the equations on the same set rectangular
coordinate system.
The Graphing Method:
1. Carefully graph each equation on the same
rectangular coordinate system.
2. If the lines intersect, determine the coordinates of
the point of intersection of the graphs. That ordered
pair is the solution of the system.
3. Check the proposed solution in each equation of
the original system.
EXAMPLE 2
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Solve the system of equations by graphing:
Objective 3: Use Graphing to Identify
Inconsistent Systems and Dependent Equations
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A system of equations that has at least one
solution, like that in Example 2, is called a
consistent system.

A system with no solution is called an
inconsistent system.
Objective 3: Use Graphing to Identify
Inconsistent Systems and Dependent Equations

In the next example, the graphs of the equations
of the system are different lines.
 We
call equations with different graphs independent
equations.
 Some equations have the same graph and are
equivalent. Because they are different forms of the
same equation, they are called dependent
equations.
Objective 3: Use Graphing to Identify
Inconsistent Systems and Dependent Equations

There are three possible outcomes when we solve a
system of two linear equations using the graphing
method:
EXAMPLE 3

Solve the system of equations by graphing:
Objective 4: Identify the Number of Solutions of
a Linear System Without Graphing

We can determine the number of solutions that
a system of two linear equations has by writing
each equation in slope–intercept form.
 If
the lines have different slopes, they intersect,
and the system has one solution. (See Example 2.)
 If the lines have the same slope and different
y-intercepts, they are parallel, and the system has
no solution. (See Example 3.)
 If the lines have the same slope and same
y-intercept, they are the same line, and the system
has infinitely many solutions. (See Example 4.)
EXAMPLE 5

Without graphing, determine the number of
solutions of: