If ⁻¹ x+ ⁻¹ y= 3 and ⁻¹ ( 1 x )- ⁻¹ ( 1 y )=0 then 2 x^2+y^2-x y=

Step-by-step solution

The second equation simplifies immediately as ⁻¹ ( 1 x ) - ⁻¹ ( 1 y ) = 0 , implying ⁻¹ ( 1 x ) = ⁻¹ ( 1 y ) , and hence 1 x = 1 y , which gives x = y . Substituting into the first equation yields 2 ⁻¹ x = 3 , so ⁻¹ x = 6 , and thus x = ( 6 ) = 1 2 . Since x = y , it follows that y = 1 2 . Evaluating the expression: 2x^2 + y^2 - xy = 2 ( 1 2 )^2 + ( 1 2 )^2 - ( 1 2 ) ( 1 2 ) = 1 2 + 1 4 - 1 4 = 1 2 . Final result: 1 2