I have a pair of odes in $x$, $y$ and $t$ along with an extra variable $z$. It looks as follows:
$\frac{dx}{dt} = \frac{1}{z}f(x,y)$, $\frac{dy}{dt} = \frac{1}{z}g(x,y)$
where $f$ and $g$ are real valued differentiable functions. Also,
$\frac{dz}{dt} = h(x,y)$
where $h$ is a real valued differentiable function.Can anyone suggest me a change of variable such that we get two self-contained odes (without any extra variable) and the system order is possibly reduced to two.
Show using the chain rule that $$\frac{dy}{dx} = \frac{g(x,y)}{f(x,y)}.$$ Now, you have a general nonlinear ode. This will give $y$ as a function of $x$. Then, $$z(t)\frac{dx}{dt} = f(x,y(x)) = h(x).$$ Separating variables, $$\frac{dx}{h(x)} = \frac{dt}{z(t)}.$$ This will give $x$ as a function of $z$, and hence $t$. $x(t) = x(z(t))$.
This answer assumes that $z$ is a known function of $t$.