  Let $\lambda_i, \mu_j$ be non-zero real numbers not all of the same sign and
let $a_i, b_k$ be non-zero integers not all of the same sign. We investigate a
mixed Diophantine system of the shape \begin{equation*} \begin{cases} \left|
\lambda_1 x_1^\theta + \cdots + \lambda_\ell x_\ell^\theta + \mu_1 y_1^\theta +
\cdots + \mu_m y_m^\theta \right| < \tau \\[10pt] a_1 x_1^d + \cdots a_\ell
x_\ell^d + b_1 z_1^d + \cdots + b_n z_n^d =0, \end{cases} \end{equation*} where
$ d\geq 2 $ is an integer, $ \theta > d+1$ is real and non-integral and $ \tau
$ is a positive real number. For such systems we obtain an asymptotic formula
for the number of positive integer solutions $(\textbf{x}, \textbf{y},
\textbf{z}) = (x_1, \ldots, z_n)$ inside a bounded box. Our approach makes use
of a two-dimensional version of the classical Hardy-Littlewood circle method
and the Davenport--Heilbronn--Freeman method. The proof involves a combination
of essentially optimal mean value estimates for the auxiliary exponential sums,
together with estimates stemming from the classical Weyl and Weyl-van der
Corput inequalities.
