Transcript 8.1 Solving Systems of Linear Equations by Graphing

8.1 Solving Systems
of Linear Equations
by Graphing
Solving Systems of Linear Equations by Graphing
A system of linear equations, often called a linear system, consists of two or
more linear equations with the same variables. Examples of systems include
2x  3y  4
3x  y  5
x  3y  4
 y  4  2x
x  y 1
y  3.
Linear systems
In the system on the right, think of y = 3 as an equation in two variables by writing it
as 0x + y = 3.
Slide 8.1-3
Objective 1
Decide whether a given ordered pair is
a solution of a system.
Slide 8.1-4
Decide whether a given ordered pair is a solution of a system.
A solution of a system of a linear equations is an ordered pair that makes
both equations true at the same time. A solution is said to satisfy the
equation.
Slide 8.1-5
CLASSROOM
EXAMPLE 1
Determining Whether an Ordered Pair is a Solution
Decide whether the ordered pair (4,−1) is a solution of each system.
Solution:
5 x  6 y  14
2x  5 y  3
 x  y  3
x  y  3
5  4   6  1  14
  4    1  3
2  4   5  1  3
20  6  14
85  3
14  14
3  3 Yes
 4    1  3
4  1  3
4  1  3
3  3
3  3
No
Slide 8.1-6
Objective 2
Solve linear systems by graphing.
Slide 8.1-7
Solve linear systems by graphing.
The set of all ordered pairs that are solutions of a system is its solution set.
One way to find the solution set of a system of two linear equations is to
graph both equations on the same axes. The graph of each line shows points
whose coordinates satisfy the equation of that line.
Any intersection point would be on both lines and would therefore be a
solution of both equations. Thus, the coordinates of any point at which the
lines intersect give a solution of the system.
Because the two different straight lines can intersect at no more then one
point, there can never be more than one solution set for such a system.
Slide 8.1-8
Solve linear systems by graphing. (cont’d)
Solving a Linear System by Graphing
Step 1: Graph each equation of the system on the same
axes.
coordinate
Step 2: Find the coordinates of the point of intersection of
if possible. This is the solution of the system.
the graphs
Step 3: Check the solution in both of the original equations. Then write the
solution set.
A difficulty with the graphing method is that it may not be possible to determine
from the graph the exact coordinates of the point that represents the solution,
particularly if those coordinates are not integers. The graphing method does,
however, show geometrically how solutions are found and is useful when
approximate answer will do.
Slide 8.1-9
CLASSROOM
EXAMPLE 2
Solving a System by Graphing
Solve the system by graphing.
5x  3 y  9
x  2y  7
Solution: {(3,2)}
Slide 8.1-10
Objective 3
Solve special systems by graphing.
Slide 8.1-11
Solve special systems by graphing.
Sometimes the graphs of the two equations in a system either do not
intersect at all or are the same line.
When a system has an infinite number of solutions, either equation of the
system could be used to write the solution set. It’s best to use the equation (in
standard form) with coefficients that are integers having no common factor
(except 1).
Slide 8.1-12
CLASSROOM
EXAMPLE 3
Solving Special Systems by Graphing
Solve each system by graphing
3x  y  4
6 x  2 y  12
Solution:

2x  5 y  8
4 x  10 y  16
 x, y  2 x  5 y  8
Slide 8.1-13
Solve special systems by graphing. (cont’d)
Three Cases for Solutions of Systems
1. The graphs intersect at exactly one point, which gives the (single) ordered
pair solution of the system. The system is consistent and the equations are
independent. See below left.
2. The graphs are parallel lines, so there is no solution and the solution set is
Ø. The system is inconsistent and the equations are independent. See
below middle.
3. The graphs are the same line. There is an infinite number of solutions, and
the solution set is written in set-builder notation as {(x,y)|_________},
where one of the equations is written after the | symbol. The system is
consistent and equations are dependent. See below right.
Slide 8.1-14
Objective 4
Identify special systems without
graphing.
Slide 8.1-15
Identify special systems without graphing.
Example 3 showed that the graphs of an inconsistent system are parallel
lines and the graphs of a system of dependent equations are the same line.
We can recognize these special kinds of systems without graphing by using
slopes.
Slide 8.1-16
CLASSROOM
EXAMPLE 4
Identifying the Three Cases by Using Slopes
Describe each system without graphing. State the number of solutions.
2x  3y  5
3y  2x  7
2
5
y 
x
3
3
2
7
y 
x
3
3
Solution:
The equations
represent parallel
lines. The system
has no solution.
x  3y  2
2 x  6 y  4
6x  y  3
2 x  y  11
1
2
y 
x
3
3
1
2
y 
x
3
3
y  6 x  3
y  2 x  11
The equations
represent the
same line. The
system has an
infinite number of
solutions.
The equations
represent lines that
are neither parallel
nor the same line.
The system has
exactly one
solution.
Slide 8.1-17