[tex] \left \{ {{3x^{2}+5xy-4y^{2}=-2} \atop {5x^{2}+9xy-2y^{2}=0}} \right. [/tex]Решите систему уравнений. Created by Freakazoid. algebra-ru

tex]Решите систему уравнений...

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[tex] \left \{ {{3x^{2}+5xy-4y^{2}=-2} \atop {5x^{2}+9xy-2y^{2}=0}} \right. [/tex]
Решите систему уравнений

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Artem112

Verified answer

$\left \{ {{3x^{2}+5xy-4y^{2}=-2} \atop {5x^{2}+9xy-2y^{2}=0}} \right. 
\\\
 \left \{ {{3x^{2}+5xy-4y^{2}=-2} \atop {5x^{2}+10xy-xy-2y^{2}=0}} \right. 
\\\
 \left \{ {{3x^{2}+5xy-4y^{2}=-2} \atop {5x(x+2y)-y(x+2y)=0}} \right. 
\\\
 \left \{ {{3x^{2}+5xy-4y^{2}=-2} \atop {(x+2y)(5x-y)=0}} \right.$

$\left \{ {{3x^{2}+5xy-4y^{2}=-2} \atop {x+2y=0}} \right. 
\\\
x=-2y
\\\
3(-2y)^2+5(-2y)y-4y^2=-2
\\\
12y^2-10y^2-4y^2=-2
\\\
y^2=1
\\\
y_1=1
\\\
y_2=-1
\\\
x_1=-2
\\\
x_2=2$

$\left \{ {{3x^{2}+5xy-4y^{2}=-2} \atop {5x-y=0}} \right. \\\ y=5x \\\ 3x^2+5x(5x)-4(5x)^2=-2 \\\ 3x^2+25x^2-100x^2=-2 \\\ x^2= \frac{1}{36} \\\ x_3= \frac{1}{6} \\\ x_4=-\frac{1}{6} \\\ y_3=\frac{5}{6} \\\ y_4=-\frac{5}{6}$

Ответ: (-2; 1); (2; -1); (1/6; 5/6); (-1/6; -5/6)

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