Deduction for joining two non linear equations

Question

I have, as dummy in math, to make the deduction for joining two nonlinear equations. I have one equation $$Y_v = 0.0000464 + x_1^{1.61} + x_2^{1.24}$$ that expresses $Y_v$ as a function of $x_1$ and $x_2$ variables. I have also another equation $$Y_v = 0.199 \cdot e^\frac{-x_3}{1.67}$$ that expresses the same $Y_v$, but now with a $x_3$ variable. For this, I need help to join $$Y_v = 0.0000464 + x_1^{1.61} + x_2^{1.24}$$ and $$Y_v = 0.199 \cdot e^\frac{-x_3}{1.67}$$ into a single equation by deduction, considering that my final objective to have $x_1$, $x_2$ and $x_3$ in the same equation.

Could any member help me? Thanks in advance!

Answer 1

$$Y_v = \beta_0 + x_1^{\beta_1} + x_2^{\beta_2}\tag 1$$ $$Y_v = \beta_0 \cdot e^\frac{-x_3}{\beta_1} \tag 2$$

Form $(1)$ or $(2)$, eliminate $\beta_0$ and replace in the other and solve for $Y_v$.

Answer 2

If you just want to eliminate $Y_v$ to get an equation in $x_1, x_2, x_3$ only, then this is simple:

$0.0000464 + x_1^{1.61} + x_2^{1.24} - 0.199 \cdot e^\frac{-x_3}{1.67} = 0$

By re-arranging this you can express $x_1$ as a function of $x_2$ and $x_3$ etc.