Answer
$$\left\{ {\left( { - \sqrt {41} ,4i} \right),\left( { - \sqrt {41} , - 4i} \right),\left( {\sqrt {41} ,4i} \right),\left( {\sqrt {41} , - 4i} \right)} \right\}$$
Work Step by Step
$$\eqalign{
& \,\,\,{x^2} + 2{y^2} = 9\,\,\,\,\,\,\,\left( {\bf{1}} \right) \cr
& \,\,\,\,\,{x^2} + {y^2} = 25\,\,\,\,\left( {\bf{2}} \right) \cr
& {\text{Multiply the equation }}\left( {\bf{2}} \right){\text{ by }} - 2{\text{ and add both equations to}} \cr
& {\text{eliminate }}{y^2} \cr
& \,\,\,\,\,\,{x^2} + 2{y^2} = 9 \cr
& \underline { - 2{x^2} - 2{y^2} = - 50} \cr
& - {x^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - 41 \cr
& \cr
& {\text{Solve the quadratic equation }} - {x^2} = - 41 \cr
& {x^2} = 41 \cr
& x = \pm \sqrt {41} \cr
& {x_1} = - \sqrt {41} ,\,\,\,{x_2} = \sqrt {41} \cr
& \cr
& {\text{From the equation }}\left( {\bf{2}} \right){\text{ we have that}} \cr
& {x^2} + {y^2} = 25 \cr
& {y^2} = 25 - {x^2} \cr
& \cr
& {\text{Substitute }}{x_1} = - \sqrt {41} {\text{ into the equation }}{y^2} = 25 - {x^2} \cr
& {y^2} = 25 - {\left( { - \sqrt {41} } \right)^2} \cr
& {y^2} = - 16 \cr
& y = \pm 4i \cr
& {\text{The first and second solutions are }}\left( { - \sqrt {41} ,4i} \right){\text{ and }}\left( { - \sqrt {41} , - 4i} \right) \cr
& \cr
& {\text{Substitute }}{x_2} = \sqrt {41} {\text{ into the equation }}{y^2} = 25 - {x^2} \cr
& {y^2} = 25 - {\left( {\sqrt {41} } \right)^2} \cr
& {y^2} = - 16 \cr
& y = \pm 4i \cr
& {\text{The third and fourth solutions are }}\left( {\sqrt {41} ,4i} \right){\text{ and }}\left( {\sqrt {41} , - 4i} \right) \cr
& \cr
& {\text{Therefore, the solution set of the system is}} \cr
& \left\{ {\left( { - \sqrt {41} ,4i} \right),\left( { - \sqrt {41} , - 4i} \right),\left( {\sqrt {41} ,4i} \right),\left( {\sqrt {41} , - 4i} \right)} \right\} \cr} $$
