Think of the number 11
Find
x
,
y
,
z
satisfying the equations
⎩
⎪
⎪
⎨
⎪
⎪
⎧
(
x
+
y
)
(
x
+
y
+
z
)
=
6
6
(
y
+
z
)
(
x
+
y
+
z
)
=
9
9
(
z
+
x
)
(
x
+
y
+
z
)
=
7
7
Submit your answer as the sum of all possible values of ( x + y + z ) .
The answer is 0.
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1 solution
Alternate solution:
Since
(
x
+
y
+
z
)
is common in all the equations,we can easy predict that
x
+
y
+
z
=
±
1
1
.
(because
±
1
1
is divisible in all the equations.So the sum of all possible values of
(
x
+
y
+
z
)
=
1
1
−
1
1
=
0
.
Adding the three equations, we get
2 ( x + y + z ) 2 = 2 4 2 or ( x + y + z ) 2 = 1 2 1 ⇒ ( x + y + z ) 2 = ± 1 1 .
Therefore the sum of all possible values of ( x + y + z ) = 1 1 − 1 1 = 0 .