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CHAPTER
1
SECTION 1
2.
(d)
1
1
111
021

21
004
1

2
000
1

3
0
0
002
5.
(a) 3
x
1
+2
x
2
=8
x
1
+5
x
2
=7
(b) 5
x
1

2
x
2
+
x
3
=3
2
x
1
+3
x
2

4
x
3
=0
(c) 2
x
1
+
x
2
+4
x
3
=

1
4
x
1

2
x
2
x
3
=4
5
x
1
x
2
+6
x
2
=

1
(d) 4
x
1

3
x
2
+
x
3
x
4
3
x
1
+
x
2

5
x
3
x
4
=5
x
1
+
x
2
x
3
x
4
5
x
1
+
x
2
x
3

2
x
4
9.
Given the system

m
1
x
1
+
x
2
=
b
1

m
2
x
1
+
x
2
=
b
2
one can eliminate the variable
x
2
by subtracting the frst row From the
second. One then obtains the equivalent system

m
1
x
1
+
x
2
=
b
1
(
m
1

m
2
)
x
1
=
b
2

b
1
1
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CHAPTER 1
(a) If
m
1
±
=
m
2
, then one can solve the second equation for
x
1
x
1
=
b
2

b
1
m
1

m
2
One can then plug this value of
x
1
into the Frst equation and solve for
x
2
. Thus, if
m
1
±
=
m
2
, there will be a unique ordered pair (
x
1
,x
2
) that
satisFes the two equations.
(b) If
m
1
=
m
2
, then the
x
1
term drops out in the second equation
0=
b
2

b
1
This is possible if and only if
b
1
=
b
2
.
(c) If
m
1
±
=
m
2
, then the two equations represent lines in the plane with
diﬀerent slopes. Two nonparallel lines intersect in a point. That point
will be the unique solution to the system. If
m
1
=
m
2
and
b
1
=
b
2
, then
both equations represent the same line and consequently every point on
that line will satisfy both equations. If
m
1
=
m
2
and
b
1
±
=
b
2
, then the
equations represent parallel lines. Since parallel lines do not intersect,
there is no point on both lines and hence no solution to the system.
10.
The system must be consistent since (0
,
0) is a solution.
11.
A linear equation in 3 unknowns represents a plane in three space. The
solution set to a 3
×
3 linear system would be the set of all points that lie
on all three planes. If the planes are parallel or one plane is parallel to the
line of intersection of the other two, then the solution set will be empty. The
three equations could represent the same plane or the three planes could
all intersect in a line. In either case the solution set will contain inFnitely
many points. If the three planes intersect in a point then the solution set
will contain only that point.
SECTION 2
2.
(b) The system is consistent with a unique solution (4
,

1).
4.
(b)
x
1
and
x
3
are lead variables and
x
2
is a free variable.
(d)
x
1
and
x
3
are lead variables and
x
2
and
x
4
are free variables.
(f)
x
2
and
x
3
are lead variables and
x
1
is a free variable.
5.
(l) The solution is (0
,

1
.
5
,

3
.
5).
6.
(c) The solution set consists of all ordered triples of the form (0
,

α,α
).
7.
A homogeneous linear equation in 3 unknowns corresponds to a plane that
passes through the origin in 3space. Two such equations would correspond
to two planes through the origin. If one equation is a multiple of the other,
then both represent the same plane through the origin and every point on
that plane will be a solution to the system. If one equation is not a multiple of
the other, then we have two distinct planes that intersect in a line through the
origin. Every point on the line of intersection will be a solution to the linear
system. So in either case the system must have inFnitely many solutions.
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 Spring '10
 wuyiling

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