Chapter 9 - Systems and Matrices - 9.5 Nonlinear Systems of Equations - 9.5 Exercises - Page 904: 41

$$\left\{ {\left( {\sqrt 5 ,0} \right),\left( {\sqrt 5 ,\sqrt 5 } \right),\left( { - \sqrt 5 ,0} \right),\left( { - \sqrt 5 , - \sqrt 5 } \right)} \right\}$$

$$\eqalign{ & \,\,\,\,{x^2} - xy + {y^2} = 5\,\,\,\,\,\,\left( {\bf{1}} \right) \cr & 2\,{x^2} + xy - {y^2} = 10\,\,\,\,\left( {\bf{2}} \right) \cr & \cr & {\text{Add both equations to eliminate the term }}xy{\text{ and }}{y^2} \cr & \,\,\,\,{x^2} - xy + {y^2} = 5 \cr & \underline {2\,{x^2} + xy - {y^2} = 10} \cr & 3{x^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 15 \cr & \cr & {\text{Solve the equation }}3{x^2} = 15 \cr & 3{x^2} = 15 \cr & {x^2} = 5 \cr & {x_1} = - \sqrt 5 {\text{ and }}{x_2} = \sqrt 5 \cr & \cr & {\text{Substitute }}{x_1}{\text{ = }} - \sqrt 5 {\text{ for }}y{\text{ into the equation }}\left( {\bf{1}} \right) \cr & {\left( { - \sqrt 5 } \right)^2} - \left( { - \sqrt 5 } \right)y + {y^2} = 5\, \cr & {\text{Solve for }}y \cr & 5 + \sqrt 5 y + {y^2} = 5\, \cr & y\left( {y + \sqrt 5 } \right) = 0 \cr & {y_1} = 0{\text{ and }}{y_2} = - \sqrt 5 \cr & \cr & {\text{The first and second solution are: }} \cr & \left( { - \sqrt 5 ,0} \right){\text{ and }}\left( { - \sqrt 5 , - \sqrt 5 } \right) \cr & \cr & {\text{Substitute }}{x_2}{\text{ = }}\sqrt 5 {\text{ for }}y{\text{ into the equation }}\left( {\bf{2}} \right) \cr & {\left( {\sqrt 5 } \right)^2} - \sqrt 5 y + {y^2} = 5\, \cr & {\text{Solve for }}y \cr & 5 - \sqrt 5 y + {y^2} = 5\, \cr & y\left( {y - \sqrt 5 } \right) = 0 \cr & {y_1} = 0{\text{ and }}{y_2} = \sqrt 5 \cr & \cr & {\text{The first and second solution are: }} \cr & \left( {\sqrt 5 ,0} \right){\text{ and }}\left( {\sqrt 5 ,\sqrt 5 } \right) \cr & \cr & {\text{Therefore, the solution set of the system is}} \cr & \left\{ {\left( {\sqrt 5 ,0} \right),\left( {\sqrt 5 ,\sqrt 5 } \right),\left( { - \sqrt 5 ,0} \right),\left( { - \sqrt 5 , - \sqrt 5 } \right)} \right\} \cr} $$