Solve the following system of equations: $\;$ $x^2 + y^2 = 25 - 2xy, \;\; y \left(x + y\right) = 10$
Given system of equations:
$x^2 + y^2 = 25 - 2xy$ $\;\;\; \cdots \; (1)$ $\;$ and $\;$ $y \left(x + y\right) = 10$ $\;\;\; \cdots \; (2)$
Equation $(1)$ can be written as
$x^2 + y^2 + 2xy = 25$
i.e. $\;$ $\left(x + y\right)^2 = 25$
i.e. $\;$ $x = y = \pm 5$
When $\;$ $x + y = + 5$ $\;\;\; \cdots \; (3)$, $\;$ equation $(2)$ becomes
$5y = 10$ $\implies$ $y = 2$
Substituting $\;$ $y = 2$ $\;$ in equation $(3)$ gives $\;\;$ $x = 5 - 2 = 3$
When $\;$ $x + y = - 5$ $\;\;\; \cdots \; (4)$, $\;$ equation $(2)$ becomes
$-5y = 10$ $\implies$ $y = -2$
Substituting $\;$ $y = -2$ $\;$ in equation $(4)$ gives $\;\;$ $x = -5 + 2 = -3$
$\therefore \;$ The solution to the given system of equations is $\;$ $\left(x, y\right) = \left\{\left(3, 2\right), \; \left(-3, -2\right) \right\}$