In this paper, by using Krasnosel'skii fixed point theorem and under suitable conditions, we present the existence of single and multiple positive solutions to the following systems$$\begin{aligned}(-1)^mu^{(2m)}&=\lambda f(t, u(t), v(t))=0,~~~~ t\in[a, b],\\(-1)^nv^{(2n)}&=\mu g(t, u(t), v(t))=0,~~~~ t\in[a, b],\\u^{(2i)}(a)&=u^{(2i)}(b)=0,~~~~0\leq i\leq m-1,\\v^{(2j)}(a)&=v^{(2j)}(b)=0,~~~~0\leq j\leq n-1,\end{aligned}$$where $\lambda, \mu>0, m,n\in \N$. We derive two explicit eigenvalue intervals of $\lambda$ and $\mu$ for the existence of at least onepositive solution and the existence of at least two positive solutions for the above higher order two-point boundary value problem.