ID: 755
x+y = 4z, x*y = 4z^2, express x-y in z
Aligned LLM's Response:
To
can
express
x
-
y
in
terms
of
z
using
the
given
equations
:
\n
\n
x
+
y
=
4
z
\n
x
\
*
y
=
4
z
^
2
\n
\n
First
,
let
'
s
solve
for
x
in
the
second
equation
:
\n
\n
x
\
*
y
=
4
z
^
2
\n
x
=
4
z
^
2
/
y
\n
\n
Now
,
substitute
this
expression
for
x
in
the
first
equation
:
\n
\n
x
+
y
=
4
z
\n
(
4
z
^
2
/
y
)
+
y
=
4
z
\n
\n
Now
,
let
'
s
solve
for
y
:
\n
\n
(
4
z
^
2
/
y
)
+
y
=
4
z
\n
4
z
^
2
/
y
+
y
=
4
z
\n
y
=
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
/
y
+
1
)
\n
\n
Now
,
we
can
substitute
this
expression
for
y
in
the
first
equation
to
express
x
in
terms
of
z
:
\n
\n
x
+
y
=
4
z
\n
x
+
((
4
z
^
2
+
4
z
)
/
(
4
z
^
2
/
y
+
1
))
=
4
z
\n
\n
Now
,
let
'
s
simplify
the
equation
:
\n
\n
x
+
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
/
y
+
1
)
=
4
z
\n
x
=
4
z
^
2
/
(
4
z
^
2
/
y
+
1
)
-
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
/
y
+
1
)
\n
\n
Now
,
let
'
s
solve
for
y
:
\n
\n
y
=
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
/
y
+
1
)
\n
\n
Now
,
substitute
this
expression
for
y
in
the
equation
above
to
express
x
in
terms
of
z
:
\n
\n
x
=
4
z
^
2
/
(
4
z
^
2
/
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
/
y
+
1
)
+
1
)
-
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
/
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
/
y
+
1
)
+
1
)
\n
\n
x
=
4
z
^
2
/
(
4
z
^
2
/
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
/
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
+
4
z
)
+
1
)
+
1
)
-
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
/
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
+
4
z
)
+
1
)
\n
\n
x
=
4
z
^
2
/
(
4
z
^
2
/
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
+
4
z
)
+
1
)
-
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
/
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
+
4
z
)
+
1
)
\n
\n
x
=
4
z
^
2
/
(
4
z
^
2
+
4
z
)
-
(
4
z
^
2
+
4
z
)
/
(
4
z
^
2
+
4
z
)
\n
\n
x
=
4
z
^
2
-
4
z
^
2
\n
\n
x
=
0
\n
\n
There
fore
,
x
-
y
=
0
in
terms
of
z
.
</s>
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