Question

Решите систему уравнений: x+xy^3=9 и xy+xy^2=6

Answer 1

9=x+xy^3 = x*(1+y^3) = x*(1+y)*(1-y + y^2),
6=xy + xy^2 = x*y*(1+y).
Разделим одно уравнение на другое:
9/6 = [ x*(1+y)*(1 - y + y^2) ]/[xy*(1+y)],
3/2 = (1 - y + y^2)/y;
3y = 2*(1-y + y^2);
3y = 2 - 2y + 2*y^2;
2*y^2 - 5y + 2 = 0;
D = 5^2 - 4*2*2 = 25 - 16 = 9 = 3^2;
y1 = (5-3)/4 = 2/4 = 1/2;
y2 = (5+3)/4 = 8/4 = 2;
x+xy^3 = 9;
x*(1+y^3) = 9;
x = 9/(1+y^3).
x1 = 9/(1+ (1/2)^3) = 9*8/(8+1) = 8; y1=1/2.
x2 = 9/(1+ 2^3) = 9/9 = 1; y2 = 2.
Ответ. (8;1/2), (1;2).