Suppose I have a parameter dependent on 2 variables
y = f(x,z)
and I have two equations defining the relation as
y = ax + b (when z is kept constant) and
y = mz³ (when x is kept constant)
How do I derive a final equation for y which is dependent on both x and z?
So we have $f(x,z) = a(z)x + b(z)$ and $f(x,z) = m(x)z^3$, hence $$ a(z)x + b(z) = m(x)z^3 \iff m(x) = \frac{a(z)}{z^3}x + \frac{b(z)}{z^3} $$ As $m$ depends only on $x$, we must have $a(z) = \alpha z^3$, $b(z) = \beta z^3$ for some $\alpha, \beta$, giving $$ f(x,z) = \alpha xz^3 + \beta z^3 $$
