Your first equation *is* an approximate solution (the coefficient of `x8`

is indistinguishable from 1) .

It turns out that your two equations are *not* equivalent :

```
sage: var("x8")
x8
sage: Sys=[1.00000000000000*x8 == 0.923879532511287,0.414213562373096*x8 == 0.382683432365089]
sage: Sys[1].solve(x8)[0].rhs()-Sys[0].solve(x8)[0].rhs()
-9631419/1809844328933473637045
sage: (Sys[1].solve(x8)[0].rhs()-Sys[0].solve(x8)[0].rhs()).n()
-5.32168366418327e-15
```

And Sage is *right* in telling you that your system has no solution.

Workarounds :

Obtain exact values for your systems' coefficients ;

solve for one equation, use the second to estimate the error, and decide if you want to accept this as an approximation. ;

treat your system as a system of redundant equations and search for the least-squares solution (left as an exercise for the reader...).

Note, by default, Sage (9.3.beta2) converts your equations' coefficients to rational approximations :

```
sage: Sys[0].solve(x8)[0].rhs()
15965419/17280845
sage: Sys[1].solve(x8)[0].rhs()
96759049255592/104731240221961
```

You can obtain floating-point solutions :

```
sage: (Sys[0].lhs()-Sys[0].rhs()).roots(ring=RR, multiplicities=False)
[0.923879532511287]
sage: (Sys[1].lhs()-Sys[1].rhs()).roots(ring=RR, multiplicities=False)
[0.923879532511283]
```

And it turns out that, even in the inexact ring `RR`

, your equations are *not* equivalent.