Prove that any of the elementary operations in Theorem 1 applied to system
(
2
)
produces an equivalent system. [Hint: To simplify this proof, represent the
i
th
equation in system
(
2
)
as
f
i
(
x
1
,
x
2
,
…
,
x
n
)
=
b
i
; so
f
i
(
x
1
,
x
2
,
…
,
x
n
)
=
a
i
1
x
1
+
a
i
2
x
2
+
⋯
+
a
i
n
x
n
for
i
=
1
,
2
,
…
,
m
. With this notation, system
(
2
)
has the form of
(
A
)
, which follows. Next, for example if a multiple of
c
times the
j
th
equation is added to the
k
th
equation, a new system of the form
(
B
)
is produced:
(
A
)
f
1
(
x
1
,
x
2
,
…
,
x
n
)
=
b
1
⋮
⋮
f
j
(
x
1
,
x
2
,
…
,
x
n
)
=
b
j
⋮
⋮
f
k
(
x
1
,
x
2
,
…
,
x
n
)
=
b
k
⋮
⋮
f
m
(
x
1
,
x
2
,
…
,
x
n
)
=
b
m
|
(
B
)
f
1
(
x
1
,
x
2
,
…
,
x
n
)
=
b
1
⋮
⋮
f
j
(
x
1
,
x
2
,
…
,
x
n
)
=
b
j
⋮
⋮
g
(
x
1
,
x
2
,
…
,
x
n
)
=
r
⋮
⋮
f
m
(
x
1
,
x
2
,
…
,
x
n
)
=
b
m
|
where
g
(
x
1
,
x
2
,
…
,
x
n
)
=
f
k
(
x
1
,
x
2
,
…
,
x
n
)
+
c
f
j
(
x
1
,
x
2
,
…
,
x
n
)
, and
r
=
b
k
+
c
b
j
. To show that the operation gives an equivalent system, show that any solution for
(
A
)
is a solution for
(
B
)
, and vice versa.]