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Baltic Way
2000 Baltic Way
17
17
Part of
2000 Baltic Way
Problems
(1)
Simultaneous equations with x,y,z,t
Source: Baltic Way 2000
12/17/2010
Find all real solutions to the following system of equations:
{
x
+
y
+
z
+
t
=
5
x
y
+
y
z
+
z
t
+
t
x
=
4
x
y
z
+
y
z
t
+
z
t
x
+
t
x
y
=
3
x
y
z
t
=
−
1
\begin{cases} x+y+z+t=5\\xy+yz+zt+tx=4\\xyz+yzt+ztx+txy=3\\xyzt=-1\end{cases}
⎩
⎨
⎧
x
+
y
+
z
+
t
=
5
x
y
+
yz
+
z
t
+
t
x
=
4
x
yz
+
yz
t
+
z
t
x
+
t
x
y
=
3
x
yz
t
=
−
1
Vieta
algebra
polynomial
quadratics
system of equations
algebra proposed