I don’t have the specific models or equations.
Here are solved forms for common model types.
If you share your exact models, I’ll tailor the steps to them.
1) Linear model: y = m x + b
1. Start with y = m x + b
2. Subtract b: y – b = m x
3. Solve for x: x = (y – b) / m (assuming m ≠ 0)
2) Quadratic model: y = a x^2 + b x + c
1. Rearrange: a x^2 + b x + (c – y) = 0
2. Discriminant: D = b^2 – 4 a (c – y)
3. Solve for x: x = [ -b ± sqrt(D) ] / (2a) (assuming a ≠ 0)
3) Exponential model: y = A e^{k x}
1. Start with y = A e^{k x}
2. Divide by A: y/A = e^{k x}
3. Take natural log: ln(y/A) = k x
4. Solve for x: x = (1/k) ln(y/A) (assuming A > 0, k ≠ 0, y > 0)
4) Power-law model: y = p x^q
1. Start with y = p x^q
2. Divide by p: y/p = x^q
3. Take q-th root: x = (y/p)^{1/q} (assuming p ≠ 0, q ≠ 0)
5) Logistic model: y = L / (1 + e^{-k (x – x0)})
1. Start with y = L / (1 + e^{-k (x – x0)})
2. Invert: 1 + e^{-k (x – x0)} = L / y
3. Subtract 1: e^{-k (x – x0)} = L/y – 1
4. Natural log: -k (x – x0) = ln(L/y – 1)
5. Solve for x: x = x0 – (1/k) ln(L/y – 1) (assuming y ∈ (0, L))
6) First-order ODE: dy/dt = r y with y(0) = y0
1. Solve differential equation: dy/y = r dt
2. Integrate: ln|y| = r t + C
3. Apply initial condition: C = ln|y0|
4. Exponentiate: y = y0 e^{r t}
5. Solve for t given y: t = (1/r) ln(y / y0) (assuming r ≠ 0)
If you provide the exact equations (models) and what you want solved for (x, t, etc.), I’ll write the precise, step-by-step solution in the same numbered, concise format.