Algebra

Find all pairs (x,y) of real numbers such that
$x + y = 10 \\ x^2 + y^2 = 56 + xy.$

$\begin{array}{|rcll|} \hline a+b &=& 10 \\ \mathbf{b} &=& \mathbf{10-a} \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline (x+y)^2 &=& x^2+2xy+y^2 \\ (x+y)^2 &=& x^2+y^2 +2xy \quad | \quad x^2 + y^2 = 56 + xy \\ 10^2 &=& 56 + xy +2xy\\ 10^2 &=& 56+3xy \\ 3xy &=& 100-56 \\ 3xy &=& 44 \quad | \quad : 3 \\ xy &=& \dfrac{44}{3} \quad | \quad y=10-x \\ x(10-x) &=& \dfrac{44}{3} \\ 10x-x^2 &=& \dfrac{44}{3} \\ \mathbf{x^2-10x+ \dfrac{44}{3}} &=& \mathbf{0} \\ \hline x &=& \dfrac{ 10\pm \sqrt{10^2 -4*\dfrac{44}{3}} }{2} \\ \\ x &=& \dfrac{ 10\pm \sqrt{4*25 -4*\dfrac{44}{3}} }{2} \\ \\ x &=& \dfrac{ 10\pm 2\sqrt{25 -\dfrac{44}{3}} }{2} \\ \\ x &=& 5\pm \sqrt{25 -\dfrac{44}{3}} \\ \\ x &=& 5\pm \sqrt{\dfrac{75-44}{3}} \\ \\ \mathbf{x} &=& \mathbf{5\pm \sqrt{\dfrac{31}{3}}} \\ \hline \end{array}$

1.

$\begin{array}{|rclrcl|} \hline \mathbf{x} &=& \mathbf{5+ \sqrt{\dfrac{31}{3}}} \\ &&&y &=& 10-a \\ &&&&=& 10-\left(5+ \sqrt{\dfrac{31}{3}}\right) \\ &&&\mathbf{y} &=& \mathbf{5- \sqrt{\dfrac{31}{3}}} \\ \hline \end{array}$

2.

$\begin{array}{|rclrcl|} \hline \mathbf{x} &=& \mathbf{5- \sqrt{\dfrac{31}{3}}} \\ &&&y &=& 10-a \\ &&&&=& 10-\left(5- \sqrt{\dfrac{31}{3}}\right) \\ &&&\mathbf{y} &=& \mathbf{5+ \sqrt{\dfrac{31}{3}}} \\ \hline \end{array}$