For a less trivial example, let us find the REAL solutions of F = G = 0, whereG:=x^6+3*x^4*y^2-(30003*x^4)/10000+3*x^2*y^4-(30003*x^2*y^2)/5000+(300060003*x^2)/100000000+y^6-(30003*y^4)/10000+(300060003*y^2)/100000000-1000300030001/1000000000000;

NiM+SSJHRzYiLDYqJEkieEdGJSIiJyIiIiomRigiIiVJInlHRiUiIiMiIiQqJEYoRiwjISYuKyQiJisrIiomRihGLkYtRixGLyomRihGLkYtRi4jRjIiJStdKiRGKEYuIyIqLisxKyQiKisrKysiKiRGLUYpRioqJEYtRixGMSokRi1GLkY5IyEuLCsuKy4rIiIuKysrKysrIkYq

F:= 27+171*y^2+66*x^2*y^2-72*x^2*y+3*x^4*y^2-12*x^4*y+3*x^ 2*y^4-24*x^2*y^3+27*x^2-108*y+9*x^4+57*y^4-136*y^3 +x^6+y^6-12*y^5;

NiM+SSJGRzYiLEIiI0YiIiIqJEkieUdGJSIiIyIkciIqJkkieEdGJUYrRipGKyIjbSomRi5GK0YqRighI3MqJkYuIiIlRipGKyIiJComRi5GM0YqRighIzcqJkYuRitGKkYzRjQqJkYuRitGKkY0ISNDKiRGLkYrRidGKiEkMyIqJEYuRjMiIioqJEYqRjMiI2QqJEYqRjQhJE8iKiRGLiIiJ0YoKiRGKkZDRigqJEYqIiImRjY=

Before we use resultants let us blindly try the numerical routine included in Maple. To do this, we enter the command p:=solve({F,G1},{x,y});

NiM+SSJwRzYiPCQvSSJ5R0YlIyImLCslIiYrKyUvSSJ4R0YlLCQtSSdSb290T2ZHNiRJKnByb3RlY3RlZEdGMkkoX3N5c2xpYkdGJTYkLCYqJEkjX1pHRiUiIiMiIiIhJioqKnpGOS9JJmxhYmVsR0YxSSVfTDEwR0YlI0Y5Ris=

And if you use evalf to numerically evaluate this result, we obtain evalf({p});

NiM8IzwkL0kieEc2IiQiKz1PLXJxISM3L0kieUdGJyQiKytdLSs1ISIq

On the other hand we can ask maple to calculate the resultant of F and G with respect to x. One gets R1:=resultant(F,G1,x);

NiM+SSNSMUc2IiokLDYqJEkieUdGJSIiKiInV0BFKiRGKSIiKSMhK2tvZnU5IiREJyokRikiIigjIi4vQkslZSdvJCInRDFSKiRGKSIiJyMhMSUzITM1S1N3YCIqRDE5VyMqJEYpIiImIyI2aiszNVtnR2hIaCIiLisrXTdHKVsqJEYpIiIlIyE6aitnNyEzNUtTaytfayIzKysrKytdN2A+KiRGKSIiJCMiPkArLzBTXSEpb1UxSyEqR2cpIjcrKysrKysrK0QxUiokRikiIiMjIUM0K18tQ0lnLFUxV05uIT1bWiIiPCsrKysrKysrKysrRGMiRikjIkc0KylHPy5rREtHQDZPQzd2PiUqKmUiQSsrKysrKysrKysrKysrK0QjIUssK08rdzB3YGNBRiFwMiVlKUgpKUg/aSMiRisrKysrKysrKysrKysrKysrKyIiIiJGSw==

This unwieldy polynomial can be simplified by typingfactor(R1);

NiMsJCokLCYhJiwrJSIiIkkieUc2IiImKyslIiM9I0YnImRvKysrKysrKysrKysrKysrKysrKysrKysrKysrKysrKysrKysrIg==

Therefore, 40001/40000 is the y-coordinate of some solution of F = G = 0 (why ?). We can now find the x-coordinate. To do this we substitute this value back into both equations, and find the common roots. We first substitute in F and factor. The Maple command isfactor(subs(y=40001/40000,F));

NiMsJCokLCYqJEkieEc2IiIiIyIrKysrKzshJioqKnoiIiIiIiQjRiwiPSsrKysrKysrKysrKyc0JQ==

factor(subs(y=40001/40000,G1));

NiMsJCokLCYqJEkieEc2IiIiIyIrKysrKzshJioqKnoiIiIiIiQjRiwiPSsrKysrKysrKysrKyc0JQ==

so (NiMtSSVzcXJ0RzYiNiMqJiImKioqeiIiIiIrKysrKzshIiI= ,NiMqJiImLCslIiIiIiYrKyUhIiI=) and (NiMsJC1JJXNxcnRHNiI2IyomIiYqKip6IiIiIisrKysrOyEiIkYs , NiMqJiImLCslIiIiIiYrKyUhIiI=) are the two REAL solutions of our system. In fact, these are the only COMPLEX solutions as well. Note that solve gave an expression for both solutions, but evalf gave (an approximation to) only one solution.