How to use elimination for five equations?

Question

$y_i = C/x_i ^q$ , i = 5

I am trying to find the average q from different points for y and x but how can i eliminate C

I tried elimination but there's 5 equation ,I'm confused can someone help me?

Answer 1

Since you have $n$ data points $(x_i,y_i)$ but only one or two parameters, this is relevant from regression. Assuming that the unknown parameters are $C$ and $q$ and the model be $$y=\frac C {x^q}$$ which is nonlinear with respect to the parameters, you can have estimates writing $$\log(y)=\log(C)-q \log(x)$$ So define first $z_i=\log(y_i)$ and $t_i=\log(x_i)$ to have $$z=\alpha +\beta t$$ which is a linear regression (easy to do). From it $C=e^\alpha$ and $q=-\beta$. With these initial parameters, you can start a nonlinear regression and get the optimal parameters which minimize the sum of squared errors.

If $C$ is fixed, then define $z_i=\log(\frac {y_i} C)$ which makes the model to become $$z=\beta t$$ and the estimate is $$-q=\beta=\frac{\sum_{i=1}^n t_i z_i}{\sum_{i=1}^n t_i^2}$$ but, again, this is an estimate and nonlinear regression should be performed since what has been measured is $y_i$ and not any of its possible transforms.