Chapter 8 Systems Of Linear Equations And Inequalities Worksheet With Answers
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Chapter 8 Systems of Linear Equations and Inequalities
S O L V I N G S Y S T E M S B Y G R A P H I N G
8.1
A N D S U B S T I T U T I O N
In Chapter 4 we studied linear equations in two variables, but we have usually con-
I n t h i s
sidered only one equation at a time. In this chapter we will see problems that in-
s e c t i o n
volve more than one equation. Any collection of two or more equations is called a
system of equations. If the equations of a system involve two variables, then the set
Solving a System by
G
of ordered pairs that satisfy all of the equations is the solution set of the system. In
Graphing
this section we solve systems of linear equations in two variables and use systems
Independent, Inconsistent,
G
to solve problems.
and Dependent Equations
Solving by Substitution
G
Solving a System by Graphing
Applications
G
Because the graph of each linear equation is a line, points that satisfy both equations
lie on both lines. For some systems these points can be found by graphing.
E X A M P L E
1
A system with only one solution
Solve the system by graphing:
y
x
2
x
y
4
Solution
First write the equations in slope-intercept form:
c a l c u l a t o r
y
x
2
y
x
4
Use the y-intercept and the slope to graph each line. The graph of the system is
c l o s e - u p
shown in Fig. 8.1. From the graph it appears that these lines intersect at (1, 3). To be
To check Example 1, graph
certain, we can check that (1, 3) satisfies both equations. Let x
1 and y
3 in
y
x
2 to get
y
x
2
1
and
3
1
2.
y
x
4.
2
Let x
1 and y
3 in x
y
4 to get
From the CALC menu, choose
intersect to have the calcu-
1
3
4.
lator locate the point of
intersection of the two lines.
Because (1, 3) satisfies both equations, the solution set to the system is (1, 3) .
After choosing intersect, you
y
must indicate which two lines
you want to intersect and
5
then guess the point of
y = x + 2
intersection.
3
?
10
1
10
10
– 3
– 1
x
1
2
3
– 1
y = –x + 4
– 2
10
I
F I G U R E 8 . 1
The graphs of the equations in the next example are parallel lines, and there is
no point of intersection.
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