Re: Eliminate

• To: mathgroup at smc.vnet.net
• Subject: [mg116176] Re: Eliminate
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Fri, 4 Feb 2011 01:38:57 -0500 (EST)

```t is already eliminated (since it isn't there).

You can eliminate x:

eqns = {x, y,
z} == {(v + w) (b^2 v^4 - a^2 v^3 w + 3 b^2 v^3 w + c^2 v^3 w -
3 a^2 v^2 w^2 + 3 b^2 v^2 w^2 + 3 c^2 v^2 w^2 - a^2 v w^3 +
b^2 v w^3 + 3 c^2 v w^3 + c^2 w^4), -w (-b^2 v^4 -
2 b^2 v^3 w - a^2 v w^3 + b^2 v w^3 + c^2 v w^3 +
c^2 w^4), -v (b^2 v^4 - a^2 v^3 w + b^2 v^3 w + c^2 v^3 w -
2 c^2 v w^3 - c^2 w^4)} // Thread;
Eliminate[eqns, x]

y == w (b^2 v^4 + 2 b^2 v^3 w + a^2 v w^3 - b^2 v w^3 - c^2 v w^3 -
c^2 w^4) &&
z == v (-b^2 v^4 + a^2 v^3 w - b^2 v^3 w - c^2 v^3 w + 2 c^2 v w^3 +
c^2 w^4)

or z:

Eliminate[eqns, z]

x == b^2 v^5 - a^2 v^4 w + 4 b^2 v^4 w + c^2 v^4 w - 4 a^2 v^3 w^2 +
6 b^2 v^3 w^2 + 4 c^2 v^3 w^2 - 4 a^2 v^2 w^3 + 4 b^2 v^2 w^3 +
6 c^2 v^2 w^3 - a^2 v w^4 + b^2 v w^4 + 4 c^2 v w^4 + c^2 w^5 &&
y == w (b^2 v^4 + 2 b^2 v^3 w + a^2 v w^3 - b^2 v w^3 - c^2 v w^3 -
c^2 w^4)

or y:

Eliminate[eqns, y]

x == b^2 v^5 - a^2 v^4 w + 4 b^2 v^4 w + c^2 v^4 w - 4 a^2 v^3 w^2 +
6 b^2 v^3 w^2 + 4 c^2 v^3 w^2 - 4 a^2 v^2 w^3 + 4 b^2 v^2 w^3 +
6 c^2 v^2 w^3 - a^2 v w^4 + b^2 v w^4 + 4 c^2 v w^4 + c^2 w^5 &&
z == v (-b^2 v^4 + a^2 v^3 w - b^2 v^3 w - c^2 v^3 w + 2 c^2 v w^3 +
c^2 w^4)

or a:

Eliminate[eqns, a]

v^2 x == 4 c^2 v^4 w^3 + 12 c^2 v^3 w^4 + 12 c^2 v^2 w^5 +
4 c^2 v w^6 - v^2 y - v^2 z - 4 v w z - 4 w^2 z && (-v + w) x ==
4 b^2 v^5 w + 8 b^2 v^4 w^2 + 4 b^2 v^3 w^3 - 4 c^2 v^3 w^3 -
8 c^2 v^2 w^4 - 4 c^2 v w^5 - 3 v y - w y + v z + 3 w z &&
v^3 y == w (b^2 v^7 + 2 b^2 v^6 w + b^2 v^5 w^2 - c^2 v^3 w^4 -
2 c^2 v^2 w^5 - c^2 v w^6 + w^2 z) &&
v x^2 + v x (2 y + 6 z) ==
64 b^2 c^2 v^6 w^5 + 144 b^2 c^2 v^5 w^6 + 16 c^4 v^5 w^6 +
96 b^2 c^2 v^4 w^7 + 96 c^4 v^4 w^7 + 16 b^2 c^2 v^3 w^8 +
240 c^4 v^3 w^8 + 256 c^4 v^2 w^9 + 96 c^4 v w^10 -
64 c^2 v^2 w^4 y - 16 c^2 v w^5 y - v y^2 - 32 b^2 v^5 w z -
80 b^2 v^4 w^2 z - 64 b^2 v^3 w^3 z + 16 c^2 v^3 w^3 z -
16 b^2 v^2 w^4 z + 16 c^2 v^2 w^4 z - 48 c^2 v w^5 z -
96 c^2 w^6 z + 26 v y z + 16 w y z - 5 v z^2 - 16 w z^2 &&
x^3 + x^2 (3 y + 3 z) +
x (3 y^2 - 42 y z + 3 z^2) == -320 b^4 c^2 v^10 w^5 -
1344 b^4 c^2 v^9 w^6 - 2176 b^4 c^2 v^8 w^7 -
1664 b^4 c^2 v^7 w^8 - 576 b^4 c^2 v^6 w^9 +
2560 b^2 c^4 v^6 w^9 + 64 c^6 v^6 w^9 - 64 b^4 c^2 v^5 w^10 +
7168 b^2 c^4 v^5 w^10 + 576 c^6 v^5 w^10 + 7552 b^2 c^4 v^4 w^11 +
2304 c^6 v^4 w^11 + 3968 b^2 c^4 v^3 w^12 + 5376 c^6 v^3 w^12 +
1152 b^2 c^4 v^2 w^13 + 5824 c^6 v^2 w^13 + 128 b^2 c^4 v w^14 +
2240 c^6 v w^14 - 1920 c^4 v^2 w^8 y - 640 c^4 v w^9 y +
320 c^2 v^2 w^3 y^2 + 64 c^2 v w^4 y^2 - y^3 + 64 b^4 v^8 w^2 z +
256 b^4 v^7 w^3 z + 256 b^4 v^6 w^4 z - 640 b^2 c^2 v^6 w^4 z -
2560 b^2 c^2 v^5 w^5 z - 64 b^4 v^4 w^6 z -
4544 b^2 c^2 v^4 w^6 z - 5056 b^2 c^2 v^3 w^7 z -
192 c^4 v^3 w^7 z - 3648 b^2 c^2 v^2 w^8 z - 704 c^4 v^2 w^8 z -
1344 b^2 c^2 v w^9 z - 1472 c^4 v w^9 z - 128 b^2 c^2 w^10 z -
2240 c^4 w^10 z + 128 b^2 v^2 w^3 y z + 448 c^2 v^2 w^3 y z +
704 c^2 v w^4 y z + 640 c^2 w^5 y z - 19 y^2 z +
64 b^2 v^4 w z^2 + 320 b^2 v^3 w^2 z^2 + 640 b^2 v^2 w^3 z^2 +
704 b^2 v w^4 z^2 + 448 b^2 w^5 z^2 + 128 c^2 w^5 z^2 - 19 y z^2 -
z^3

Some variables probably cannot be eliminated.

Bobby

On Thu, 03 Feb 2011 04:29:59 -0600, Francisco Javier Garc=EDa Capit=E1n
<garciacapitan at gmail.com> wrote:

> Dear friends, how can I eliminate t from the equations
>
> {x,y,z}=={(v + w) (b^2 v^4 - a^2 v^3 w + 3 b^2 v^3 w + c^2 v^3 w -  3 a^2
> v^2 w^2 + 3 b^2 v^2 w^2 + 3 c^2 v^2 w^2 - a^2 v w^3 +
>     b^2 v w^3 + 3 c^2 v w^3 + c^2 w^4),
>  -w (-b^2 v^4 - 2 b^2 v^3 w - a^2 v w^3 + b^2 v w^3 + c^2 v w^3 + c^2
> w^4),
> -v (b^2 v^4 - a^2 v^3 w + b^2 v^3 w + c^2 v^3 w - 2 c^2 v w^3 - c^2 w^4)}
>
> Many thanks
>
> --
> ---
> Francisco Javier Garc==EDa Capit==E1n
> http://garciacapitan.auna.com
>

--
DrMajorBob at yahoo.com

```

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