Układy równań

[aguska9923]

\(\begin{cases} x+xy+y=7\\ x^2+xy+y^2=13\\ \end{cases}\)
\(\begin{cases}x^2-y=3\\ y^2-x=3\\ \end{cases}\)

[ewelawwy]

1.
\(x(1+y)=7-y\\
x=\frac{7-y}{1+y},\ y\neq -1\\
\left(\frac{7-y}{1+y}\right)^2+y\cdot \frac{7-y}{1+y}+y^2=13\\
\frac{49-14y+y^2}{1+2y+y^2}+\frac{7y-y^2}{1+y}+y^2=13/\cdot (1+y)^2\\
49-14y+y^2+(7y-y^2)(1+y)+y^2(1+2y+y^2)=13(1+2y+y^2)\\
49-14y+y^2+7y-y^2+7y^2-y^3+y^2+2y^3+y^4-13-26y-13y^2=0\\
y^4+y^3-5y^2-33y+36=0\\
(y-1)(y-3)(y^2+5y+12)=0\\
y=1\ \vee\ y=3\\
x=3\ \vee\ x=1\)

\(\{x=1\\y=3\)\(\ \vee\\)\(\{x=3\\y=1\)

[ewelawwy]

2.
\(y=x^2-3\\
(x^2-3)^2-x=3\\
x^4-6x^2+9-x-3=0\\
x^4-6x^2-x+6=0\\
(x-1)(x+2)(x^2-x-3)=0\\
x=1\ \vee \ x=-2\ \vee\ x=\frac{1-\sqrt{13}}2\ \vee \ x=\frac{1+\sqrt{13}}2\\
y=-2\ \vee\ y=1\ \vee \ y=\left(\frac{1-\sqrt{13}}2\right)^2-3\ \vee\ y=\left(\frac{1+\sqrt{13}}2\right)^2-3\\
y=-2\ \vee\ y=1\ \vee \ y=\frac{1-\sqrt{13}}2\ \vee\ y=\frac{1+\sqrt{13}}2\)

\(\{x=1\\ y=-2\)\(\ \vee\\)\(\{x=-2\\y=1\)\(\ \vee\\)\(\{x=\frac{1-\sqrt{13}}2\\y=\frac{1-\sqrt{13}}2\)\(\ \vee\\)\(\{x=\frac{1+\sqrt{13}}2\\y=\frac{1+\sqrt{13}}2\)

[aguska9923]

Dziękuję bardzo!!