# Automatic Threading of Equations

This little utility was originally developer for DMUG,
the German Mathematica User Group.
It answers a frequently asked question about the manipulation of
equations. Mathematica does not "thread" arithmetic and other functions
over equations, so the following naive approach to solving equations
by rearranging terms does not work:
```
In[1]:= 1 + x == 2
In[2]:= %-1
Out[2]= -1 + (1 + x == 2)
```

To make it work, the symbol `Equal`

(the head of an equation) should
behave like List with respect to listable functions. Just as
`{a, b} + 1`

turns into `{a+1, b+1}`

, one may want `(a==b) + 1`

to turn into
`a+1 == b+1`

. This can be achieved explicitly with `Thread`

:
```
In[3]:= Thread[(a==b) + 1, Equal]
Out[3]= 1 + a == 1 + b
```

The automatic transformation of `f[a, b, c, ...]`

into `Thread[f[a, b, c, ...]]`

should happen whenever `f`

has the attribute `Listable`

and at least one
of the arguments a, b, ... has head `Equal`

. This definition is essentially
what is needed:
```
Equal/: lhs:f_Symbol?listableQ[___, _Equal, ___] := Thread[ Unevaluated[lhs], Equal ]
listableQ[f_] := MemberQ[Attributes[f], Listable]
```

The use of `Unevaluated`

prevents an infinite recursion. Together with
the necessary framework, the little package
EqualThread.m implements this functionality.
Now, you can solve equation as you did in school:

read the package:

```
In[1]:= Needs["EqualThread`"]
```

the equation, to be solved for x:
```
In[7]:= a == b Log[2 x]
```

divide by b:
```
In[8]:= %/b
a
Out[8]= - == Log[2 x]
b
```

exponentiate:
```
In[9]:= Exp[%]
a/b
Out[9]= E == 2 x
```

divide by 2:
```
In[10]:= %/2
a/b
E
Out[10]= ---- == x
2
```

You can also add equations, etc.:
```
In[11]:= (a==b) + (c==d)
Out[11]= a + c == b + d
```

## Notebook and Package Copyright

© 1997 by MathConsult Dr. R. Mäder

Permission is granted to use and distribute this file for any purpose
except for inclusion in commercial software or program collections.
This copyright notice must remain intact.