The Philip's equation in order to evaluate the cumulative infiltration of water into soil is$z=s\sqrt t+kt$. For the data provided in the following table I want to find the coefficients $s$ and $k$,
$$\begin{array}{|c|c|} \hline t (\text{min})& z (\text{cm}) \\ \hline 5&2 \\ \hline 15& 4.5\\ \hline 45& 9\\ \hline 75& 13 \\ \hline 110& 17.5\\ \hline \end{array}$$ I know that we can approximate the coefficients by using Least Square Method,
Let $r_i=s\sqrt{t_i}+kt_i-z_i$ be the error of each data and we want to minimize $Q=\sum_{i=1}^5r_i^2$, hence we should have $\dfrac{\partial Q}{\partial s}=0$ and $\dfrac{\partial Q}{\partial k}=0$,
$$\dfrac{\partial Q}{\partial s}=0 \Rightarrow \dfrac{\partial Q}{\partial s}=\dfrac{\partial Q}{\partial r_i}\times \dfrac{\partial r_i}{\partial s}=2\sum_{i=1}^5r_i \times t_i^{\frac12}=0 \Rightarrow s\sum_{i=1}^5t_i+k\sum_{i=1}^5t_i^{\tfrac32}-\sum_{i=1}^5 z_it_i^{\tfrac12}=0$$
$$\dfrac{\partial Q}{\partial k}=0 \Rightarrow \dfrac{\partial Q}{\partial k}=\dfrac{\partial Q}{\partial r_i}\times \dfrac{\partial r_i}{\partial k}=2\sum_{i=1}^5r_i \times t_i=0 \Rightarrow s\sum_{i=1}^5t_i^{\tfrac32}+k\sum_{i=1}^5t_i^{2}-\sum_{i=1}^5 z_it_i=0$$ And after evaluating the above sums we get a system of linear equations in $s$ and $k$ that can be solved easily.
I'm wondering is it possible to solve this problem by hand (without using softwares like Excel) with other approaches? For example I've heard that it can be solved by matrix calculations but I'm not sure how to do it.