Assign the results from a Solve to variable(s)

Question

I understand Mathematica can't assign the results of a Solve to the unknowns because there may be more than 1 solution. How can I assign the 4 values of following result to variables?

Answer 1

You can do this :

s = Solve[y^2 == 13 x + 17 && y == 193 x + 29, {x, y}];
xx = s[[All, 1, 2]];
yy = s[[All, 2, 2]];

Now you can access solutions, this way xx[[1]], yy[[2]].

If you prefer to collect solutions in Array, there is another way :

X = Array[ x, {Length@s}];
Y = Array[ y, {Length@s}];
x[k_] /; MemberQ[ Range[ Length @ s], k] := s[[k, 1, 2]]
y[k_] /; MemberQ[ Range[ Length @ s], k] := s[[k, 2, 2]]

now X is equivalent to s[[All, 1, 2]], while Y to s[[All, 2, 2]], e.g. :

X[[1]] == x[1]
Y == s[[All, 2, 2]]

True
True

You do not have to use or even to define X and Y arrays, e.g.

{x[1], y[1]}

{(-11181 - Sqrt[2242057])/74498, 1/386 (13 - Sqrt[2242057])}

We've used Condition i.e. /; to assure definitions of x[i], y[i] only for i in an appropriate range determined by Length @ s, i.e. number of solutions.

Answer 2

Usually you don't want to actually assign values to x and y, and you would use replacement rules instead:

sols = Solve[y^2 == 13 x + 17 && y == 193 x + 29, {x, y}];

{x, y} /. sols[[1]]

or for the second solution:

{x, y} /. sols[[2]]

---

If you really want to assign values to x and y globally, you could use:

Set @@@ sols[[1]]

but you must clear x and y before using another set:

Clear[x, y]
Set @@@ sols[[2]]

---

If you want to assign values to x and y within a Block you could do something like this:

Hold @@ {sols[[2]]} /. Rule -> Set /. _[vars_] :>
  Block[vars,
   Sin[x] + Sqrt[y] // N
  ]

This uses what I am calling the injector pattern to get the values into Block in the right syntax without it prematurely evaluating.

---

Related questions:

Getting rid of the “x ->” in FindInstance results

Using the output of Solve

Answer 3

If you really wish to assign solutions to variables, you can do something like this:

In[1]:= ClearAll[Subscript]
sols=Solve[y^2==13x+17&&y==193x+29,{x,y}];
i=0;
sols/.{r__Rule}:>Set@@@({r}/.var:x|y->Subscript[var,++i]);
Subscript//Definition

Out[5]=

Subscript[x,1]=(-11181-Sqrt[2242057])/74498
Subscript[x,2]=(-11181+Sqrt[2242057])/74498
Subscript[y,1]=1/386(13-Sqrt[2242057])
Subscript[y,2]=1/386 (13+Sqrt[2242057])

Then you can use the solutions for demonstration purposes: