How to express a variable (function) in terms of specific variables?

I wish to return r1 as a function of all the symbolic variables under syms besides the [C_S, C_CS, C_BS]. But I am not exactly sure of how I should proceed.

The code below returns 4 zero solutions, which is technically correct. But the thing is that r1 is not 0, it is a function, and thus I need a function in all the symbolic variables with known IVPs.

syms C_S C_CS C_BS C_C C_P C_B r1 r2 r3 k_A k_A1 k_S k_S1 k_D k_D1

Ct = C_S + C_CS + C_BS

func1 = k_A*C_C*C_S - k_A1*C_CS
func2 = k_S*C_CS - k_S1*C_BS*C_P
func3 = k_D*C_BS - k_D1*C_B*C_S

Eq1 = [func1 == r1,  func2 == 0,  func3 == 0, Ct]

KA = k_A/k_A1
KS = k_S/k_S1
KD = k_A/k_D1

[x, y, z, w] = solve(Eq1, [r1 C_S C_CS C_BS])

[Paul]

The problem defines four equations with four variables to solve for. According to solve() there is only one solution subject to a constraint. Can you show or explain mathematically, i.e., without resorting to code, what the goal is?

syms C_S C_CS C_BS C_C C_P C_B r1 r2 r3 k_A k_A1 k_S k_S1 k_D k_D1
Ct = C_S + C_CS + C_BS;
func1 = k_A*C_C*C_S - k_A1*C_CS;
func2 = k_S*C_CS - k_S1*C_BS*C_P;
func3 = k_D*C_BS - k_D1*C_B*C_S;
Eq1 = [func1 == r1,  func2 == 0,  func3 == 0, Ct];
sol = solve(Eq1, [r1 C_S C_CS C_BS],'ReturnConditions',true)

sol = struct with fields:
            r1: 0
           C_S: 0
          C_CS: 0
          C_BS: 0
    parameters: [1×0 sym]
    conditions: k_D*k_S + C_B*k_D1*k_S + C_B*C_P*k_D1*k_S1 ~= 0 & C_B ~= 0 & k_D1 ~= 0

[Andrian Mirza]

So, in the picture below it is solved by hand, I solved for r1, the overbar stays for S in the index in the above code. So we have three rate expressions, from which r2 and r3 are equal to 0. And the goal is to express r1 as a function of all but [C_CS, C_S C_BS]. I hope that this is a clear response from my side.

Answer 1

[Paul]

syms C_S C_CS C_BS C_C C_P C_B r1 r2 r3 k_A k_A1 k_S k_S1 k_D k_D1
Ct = C_S + C_CS + C_BS

Ct = 

r1 = k_A*C_C*C_S - k_A1*C_CS;
r2 = k_S*C_CS - k_S1*C_BS*C_P;
r3 = k_D*C_BS - k_D1*C_B*C_S;
sol = solve([r2==0,r3==0],[C_CS C_S],'ReturnConditions',true)

sol = struct with fields:
          C_CS: (C_BS*C_P*k_S1)/k_S
           C_S: (C_BS*k_D)/(C_B*k_D1)
    parameters: [1×0 sym]
    conditions: C_B ~= 0 & k_D1 ~= 0 & k_S ~= 0

r1 = subs(r1,[C_CS C_S],[sol.C_CS sol.C_S])

r1 = 

[num,den]=numden(r1)

num = 

den = 

r1 = num/den

r1 =