How to solve 3 interconnected equations?

I have a vecor R with 10 values

M = S * 3 and S = R+M

how do I connect the vector with the 2 equations in order to receive a vector M with 10 values and a vector S with 10 values?

Would a for-loop be a viable solution?

Answer 1

[Ameer Hamza]

If you have symbolic toolbox, you can use solve()

syms M S R
Rv = rand(10, 1);

eq1 = M == S*3;
eq2 = S == R+M;

sol = solve([eq1; eq2], [M S]);
M_sol = subs(sol.M, Rv);
S_sol = subs(sol.S, Rv);

Result

>> M_sol
M_sol =
-14923794479723871/18014398509481984
  -4255113867394491/4503599627370496
   -108056045000625/2251799813685248
-16610538751653585/18014398509481984
 -9792936758686299/18014398509481984
 -1338449427099159/18014398509481984
  -1653620415623073/2251799813685248
 -5201938487820639/18014398509481984
  -1662959758749849/9007199254740992
   -694097603644977/2251799813685248
>> S_sol
S_sol =
-4974598159907957/18014398509481984
 -1418371289131497/4503599627370496
   -36018681666875/2251799813685248
-5536846250551195/18014398509481984
-3264312252895433/18014398509481984
 -446149809033053/18014398509481984
  -551206805207691/2251799813685248
-1733979495940213/18014398509481984
  -554319919583283/9007199254740992
  -231365867881659/2251799813685248

If you want floating-point answer

M_sol = double(M_sol);
S_sol = double(S_sol);

Reault

>> M_sol
M_sol =
   -0.2198
   -0.2836
   -0.0640
   -0.9528
   -0.4228
   -0.8079
   -1.0427
   -0.7487
   -0.8037
   -0.6678
>> S_sol
S_sol =
   -0.0733
   -0.0945
   -0.0213
   -0.3176
   -0.1409
   -0.2693
   -0.3476
   -0.2496
   -0.2679
   -0.2226

Alternatively use fsolve():

R = rand(10, 1);

eq = @(M,S) [M-S*3; S-(R+M)];
sol = fsolve(@(x) eq(x(1:10),x(11:20)), zeros(20, 1));

M_sol = sol(1:10);
S_sol = sol(11:20);

[Lu Da Silva]

Can I solve it without the use of any extra tools?

[Ameer Hamza]

For that, you will need to first solve the equations by hand. You will get

S = -R/2

M = -3R/2

and then use MATLAB

R = rand(10, 1);

S = -R/2
M = -3*R/2