How to define a series of substituting rules?

Question

suppose I have a long list of many variables: alist = { u, v, w, ... }. At some point, I have another combination of variables depending on { u, v, w, etc.},e.g., alpha = 2 * u + v beta = v + w^2. And now I want to substitute u, v, and w with the following expressions: u = 3 - x + y; v = Exp[y] + z; etc., where the right hand side ( say 3 - x + y ) is contained in another list of expressions, say blist[[1]] = 3 - x + y.

For the task stated above, I naively input alist[[i]] = blist[[i]], but obviously this will do the job. what it merely does is to assign 3 - x + y to alist[[1]] ( when i == 1 ).

I want to know if it is possible to treat alist[[1]] = 3 - x + y as a pattern rule: u = 3 - x + y rather than assign 3 - x + y to alist. Thank you !

array = {u, v, w}

Out[362]= {a, b, c}

alpha = 2 * u + v

array[[1]] = 2 x + y

In[367]:= u

u (* meant to be 2 u + y *)

In[368]:= alpha

alpha (* meant to be  4 x + 2 y + v *)

```

Answer 1

As pointed out in the comments by Alan, Rule and ReplaceAll is what you need.

An example

ReplaceAll[{Rule[u, x + y], Rule[v, x - y]}][u + v]
(* 2 x *)

which can be written more succinctly as

u + v /. {u -> x + y, v -> x - y}
(* 2 x *)

In the case that you have your alist and blist, a list of rules can be created as follows

alist = {u, v};
blist = {x + y, x - y};
rules = Thread[Rule[alist, blist]]
(* {u -> x + y, v -> x - y} *)

If you have an expression expr in terms of "a" variables, you can then replace them via

expr /. rules

Answer 2

I had some trouble understanding your question, but I think the following can be adapted to do what you want.

left = {u, v, w}
right = {3 - x + y, Exp[y] + z, etc}
rules = Thread[Rule[left, right]]
alpha = 2*u + v
beta = v + w^2
alpha /. rules
beta /. rules