1 | initial version |

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.

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.

2 | No.2 Revision |

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.~~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.

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.