Wrong solution using Cramer's rule? I have tried to solve the following systems of equations: 
$$
\left\{
  \begin{array}{rrrcl}
     x &  + y &  + z &=& 1 \\
    2x & + 5y & + 2z &=& 4 \\
    4x & + 2y & + 3z &=& 5
  \end{array}
\right.
$$
The determinant of the matrix of the coefficients is $3$ and the other determinants are $-8$, $-2$, $7$. So the solutions are 
$$
  x = - \frac{8}{3} \,,
  \quad
  y = - \frac{2}{3} \,,
  \quad
  z = \frac{7}{3} \,.
$$
However, these values doesn't satisfy the system. Furthermore, Wolfram Alpha says that the solutions are $x = 8/3, y = 2/3, z = -7/3$. The roots are the same in numerical value, but with opposite sign. What is the error here?

At least$$\Delta=1\cdot5\cdot3+1\cdot2\cdot2+4\cdot1\cdot2-4\cdot5\cdot1-1\cdot2\cdot2-3\cdot2\cdot1=-3$$
$$\Delta_x=1\cdot5\cdot3+1\cdot2\cdot5+4\cdot1\cdot2-5\cdot5\cdot1-1\cdot2\cdot2-3\cdot4\cdot1=-8$$
$$\Delta_y=1\cdot4\cdot3+1\cdot4\cdot2+5\cdot1\cdot2-4\cdot4\cdot1-1\cdot2\cdot5-3\cdot2\cdot1=-2$$ and
$$\Delta_z=1\cdot5\cdot5+1\cdot2\cdot2+4\cdot1\cdot4-4\cdot5\cdot1-1\cdot2\cdot4-5\cdot2\cdot1=7.$$
