Answer
$$\left\{ {\left( {0,0} \right)} \right\}$$
Work Step by Step
$$\eqalign{
& \,\,\,5{x^2} - {y^2} = 0\,\,\,\,\,\left( {\bf{1}} \right) \cr
& 3\,{x^2} + 4{y^2} = 0\,\,\,\,\left( {\bf{2}} \right) \cr
& {\text{Multiply the equation }}\left( {\bf{1}} \right){\text{ by 4 and Add both equations to }} \cr
& {\text{eliminate }}{y^2} \cr
& \,20{x^2} - 4{y^2} = 0 \cr
& \,\,\underline {3\,{x^2} + \,\,\,4{y^2} = 0} \cr
& \,\,\,\,\,\,23{x^2}\,\,\,\,\,\,\,\,\,\, = 0 \cr
& \cr
& {\text{Solve the quadratic equation 23}}{x^2} = 0 \cr
& 23{x^2} = 0 \cr
& x = 0 \cr
& \cr
& {\text{From the equation }}\left( {\bf{1}} \right){\text{ we have that}} \cr
& \,5{x^2} - {y^2} = 0 \cr
& {y^2} = 5{x^2} \cr
& \cr
& {\text{Substitute }}x = 0{\text{ into the equation }}{y^2} = 5{x^2} \cr
& {y^2} = 5{\left( 0 \right)^2} \cr
& y = 0 \cr
& \cr
& {\text{Therefore, the solution set of the system is}} \cr
& \left\{ {\left( {0,0} \right)} \right\} \cr} $$
