Question

(x*3 + y*3 = 2,xy(x + y) = 2;​

Answer 1

$\left\{\begin{array}{l}x^3+y^3=2\\xy(x+y)=2\end{array}\right\ \ \left\{\begin{array}{l}x^3+y^3=2\\x^2y+xy^2=2\ |\cdot 3\end{array}\right\ \oplus \ \left\{\begin{array}{l}x^2+3x^2y+3xy^2+y^3=8\\x^2y+xy^2=2\end{array}\right$

$\left\{\begin{array}{l}(x+y)^3=8\\xy(x+y)=2\end{array}\right\ \ \left\{\begin{array}{l}x+y=2\\xy\cdot 2=2\end{array}\right\ \ \left\{\begin{array}{l}x+y=2\\xy=1\end{array}\right\ \ \left\{\begin{array}{l}y=2-x\\x(2-x)=1\end{array}\right$

$\left\{\begin{array}{l}y=2-x\\2x-x^2=1\end{array}\right\ \ \left\{\begin{array}{l}y=2-x\\x^2-2x+1=0\end{array}\right\ \ \left\{\begin{array}{l}y=2-x\\(x-1)^2=0\end{array}\right\ \ \left\{\begin{array}{l}y=2-1=1\\x=1\end{array}\right\\\\\\Otvet:\ \ (\, 1\, ;\, 1\, )\ .$

Answer 2

To solve the system of equations:

1. x^3 + y^3 = 2
2. xy(x + y) = 2

You can use a substitution method. Let's solve it step by step:

First, let's solve equation (2) for one variable, say x:

xy(x + y) = 2

Divide both sides by y:

x(x + y) = 2/y

Now, let's express x in terms of y:

x(x + y) = 2/y x^2 + xy = 2/y

Now, rearrange the equation:

x^2 = (2/y) - xy

Next, substitute this expression for x^2 into equation (1):

x^3 + y^3 = 2

[(2/y) - xy]^3 + y^3 = 2

Now, let's solve for y. This equation may get a bit complicated, so it's best to use numerical methods or a computer algebra system to find an approximate solution. Let's call this equation (3).

Once you have found the value of y, you can substitute it back into equation (2) to solve for x:

xy(x + y) = 2

x(y + y) = 2

x(2y) = 2

x = 2/(2y)

Simplify:

x = 1/y

So, you have found the values of x and y. Remember that equation (3) will likely require numerical methods to find an approximate solution, as it involves a cubic polynomial.