Question

Розвяжіть систему рівннянь x+y+xy=11; x²y+xy²= 30Даю 70 балів !!!!! срочноооооо​

Answer 1

Ответ:

Объяснение:

$\displaystyle\\\left \{ {{x+y+xy=11} \atop {x^2y+xy^2=30}} \right. ;\left \{ {{x+y+xy=11} \atop {xy(x+y)=30}} \right.\\\\a=x+y;~b=xy\\\\\left \{ {{a+b=11} \atop {ab=30}} \right. ;\left \{ {{b=11-a} \atop {a(11-a)=30}} \right.; \left \{ {{b=11-a} \atop {a^2-11a+30=0} \right.$

по теореме Виета корни второго уравнения

             $a_1=6;~~~a_2=5$

тогда   $b_1=11-6=5;~~~b_2=11-5=6$

обратная замена

$1)~\displaystyle\left \{ {{x+y=6} \atop {xy=5}} \right. ;\left \{ {{y=6-x} \atop {x(6-x)=5}} \right. ;\left \{ {{y=6-x} \atop {x^2-6x+5=0}} \right.\\\\po~Vieta~x_1=1;x_2=5\\~~~~~~~~~~~~~~y_1=5;y_2=1\\\\\\2)~\displaystyle\left \{ {{x+y=5} \atop {xy=6}} \right. ;\left \{ {{y=5-x} \atop {x(5-x)=6}} \right. ;\left \{ {{y=5-x} \atop {x^2-5x+6=0}} \right.\\\\po~Vieta~x_3=2;x_4=3\\~~~~~~~~~~~~~~y_3=3;y_4=2\\\\Otvet:(1;5)~(2;3)~(3;2)~(5;1)$

Answer 2

Solving the System of Equations

To solve the system of equations x + y + xy = 11 and x²y + xy² = 30, we can use a substitution method. Let's solve it step by step.

First, let's solve for one variable in terms of the other using the first equation.

From the equation x + y + xy = 11, we can express y in terms of x as follows: y = (11 - x) / (1 + x)

Now, we can substitute this expression for y into the second equation x²y + xy² = 30.

Substituting y into the second equation gives us: x²((11 - x) / (1 + x)) + x((11 - x) / (1 + x))² = 30

Solving this equation will give us the values of x and y that satisfy both equations.

Let's proceed with solving this equation.