For two equations, one cannot simply combine them into one and say the original system is equivalent to the result. One must keep at least one of the original two equations.
For your systems, each system happens to have at least one equation which is linear in one of the two variables. In such a case you can solve the one equation for one of the two variables, and substitute the result in the other equation. Then you have two equations, one of which has one of the original variables, say $y$, on the left and does not have $y$ on the right, and the other equation is all in terms of the other variable $x$. Then if you can solve the one variable equation for $x$ these results may be put into the "solved for $y$" equation to get the corresponding $y$ values for each $x$ obtained.
Added note: when the expression in the "solved for" variable has a denominator, one has to separately consider the case in which that denominator is zero. For example solving the first equation of your first system for $y$ gives $y=(3x-x^2)/x$, so that the possibility $x=0$ has to be treated separately, and then the substitution may be used as above suggested, in this case (once $x=0$ already treated) the substitution is $y=3-x$.