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Re: Solve without vars

• To: mathgroup at smc.vnet.net
• Subject: [mg130379] Re: Solve without vars
• From: Andrzej Kozlowski <akozlowski at gmail.com>
• Date: Sat, 6 Apr 2013 05:11:52 -0400 (EDT)
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• References: <kjipr5$3kq$1@smc.vnet.net> <20130405023336.88DA36B37@smc.vnet.net>

Yes, giving an empty list of variables is the same as giving no 
variables at all, and no, Solve does not assume that all symbols are 
variables. It assumes that y and b are variables and a and x parameters. 
The reason why it makes these choices seems a little subtle.


The clearest indication that Solve is not  assuming that all symbols are 
variables is the fact that it does not produce the message that it 
produces here:

Solve[{b*x + y == 7 && a*x - y == 1}, {x, y, a, b}]
Solve::svars:Equations may not give solutions for all "solve" variables. 
>>
{{y -> -1 + a*x, b -> (8 - a*x)/x}}

In fact, what Solve seems to do is something like the following.

First y is expressed in terms of x and b as y = 7 - b x. The reason 
why Solve choses to treat y  in this way (as a "variable" rather than a 
"parameter") by doing so it does not have to make any assumptions at all 
about the other symbols. If it chose x as a "variable" (meaning, that it 
would have tried to express x in terms of y and a) it would have to make 
the assumption that a is non-zero and it wants to make as few 
assumptions as possible. Similarly, if it chose a as a variable, it 
would have to assume that x is non-zero. The choice of you is the only 
one that allows it to make no assumptions at this stage.

However, as a result of this choice, Solve is now "forced" to treat a 
and x as "parameters" (since they were used in the solution for y).
 Now, by substituting for y into the second equation, Solve gets an 
equation involving a, b, and x. However, since a and x  are now being 
treated as "parameters", b has to became a "variable" and the solution 
obtained turns out to be exactly the same as given by:

Solve[{x + b*y == 7 && a*x - y == 1}, {y, b}]

{{y -> -1 + a*x, b -> (7 - x)/(-1 + a*x)}}

You can easily check that this behaviour is not dependent on the symbols 
used for the variables - only on their role in the equations.

Next, something that seems to me more interesting. If you change these 
equations somewhat you will see behaviour that may appear even more 
surprising.

Solve[{x + b*y == 7 && a*x - y == 1}]

{{a -> (1 + y)/x, b -> (7 - x)/y}, {a -> 1/7, y -> 0, x -> 7}, {b -> -7, y -> -1, x -> 0}}

Here Solve returned a list of solutions rather than one solution. The 
first is the same one it returns when called with variable specification 
{a,b}:

Solve[{x + b*y == 7 && a*x - y == 1}, {a, b}]

{{a -> (1 + y)/x, b -> (7 - x)/y}}

This solution of course requires assumptions on the "parameters" x,y 
i.e. x!=0 and y!=0. In general, when called with specific variables, 
Solve by default does not return solutions that involve assumptions on 
parameters.  However the assumption that a parameter is non-zero is 
normally not excluded, in other words, it is not really considered "an 
assumption".  But when called without specified variables Solve actually 
does something that it normally does not do : it actually considers the 
special cases x==0 and y==0 separately and returns two 
additional solutions.

Note how similar the result of Solve is to the one given by Reduce:

Solve[{x + b*y == 7 && a*x - y == 1}]

{{a -> (1 + y)/x, b -> (7 - x)/y}, {a -> 1/7, y -> 0, x -> 7}, {b -> -7, y -> -1, x -> 0}}

Reduce[{x + b*y == 7 && a*x - y == 1}]
(y == -1 && x == 0 && b == -7) || (y == 0 && x == 7 =
&& a == 1/7) ||
  (y != 0 && b == (7 - x)/y && x != 0 && a == (1 + y)/x)

All this is completely undocumented and in fact even appears to 
contradict the documentation, since Solve does appear to be actually 
returning solutions that require parameters to have special values. Of 
course only stated that this is not done when variables are explicitly  
specified. Since we do not specify variables we are in undocumented 
territory and Solve is not obliged to follow the behavior documented for 
the case of specified variables.

Andrzej Kozlowski





On 5 Apr 2013, at 11:33, debguy <johnandsara2 at cox.net> wrote:

> test if these are the same (all Symbols assumed variables, i think):
>
> Solve[a x + y == 7 && b x - y == 1, {}]
>
> Solve[a x + y == 7 && b x - y == 1]

> On 4 Apr 2013, at 11:53, juchheissassa at gmail.com wrote:
>
>> I happened to hit on the following curious thing:
>>
>> If I use Solve with {} instead of vars, I actually get a solution.
>>
>> Is this documented anywhere? What are the rules?
>>
>> e.g.:
>>
>> Solve[a x + y == 7 && b x - y == 1, {}] yields {{y -> 7 - a x, b -> (8 - a x)/x}}
>>
>> while
>>
>> Solve[a x + y == 7 && b x - y == 1, {x,y}] yields {{x -> 8/(a + b), y -> -((a - 7 b)/(a + b))}}
>>
>
>>
>

• References:
  • Re: Solve without vars
    • From: debguy <johnandsara2@cox.net>