A consumer has utility function $U(x,y)=(x−2)y$, where $x≥2$ and $y≥0$. The price of $x$ is $P_x$, the price of $y$ is $P_y$ and the consumer's income is $I>2P_x$. ($x$ and $y$ do not have to be integers.)
I ended up with the demand functions for $x$ and $y$ to be $\frac I {2P_x}+1$ and $\frac I {2P_y}−\frac {P_x}{P_y}$ respectively.
The question then gives $P_x=1$ and $P_y=2$ and asks me to draw the Engel curve of $x$.
Assuming my demand functions are correct (I am still new to the concept of utility maximisation and derivation of demand functions), I have found three points to plot the curve: $(3,4)$, $(4,6)$ and $(5,8)$. This would result in a linear Engel curve.
I would like to know if I have gone wrong anywhere along the way, as I have always thought Engel curves to be, as the name suggests, curves. If my answer is correct, may I know what kind of good would have a linear Engel curve?
For example, I do know that Engel curves have a positive and negative gradient for normal and inferior goods respectively. Moreover, if the curve tends towards the x or y-axis, the good is a necessity and a luxury respectively.
If I could get examples for linear Engel curves with positive and negative gradients respectively, that will be great!