Solve the system by the method of elimination. { 8 x 2 − 2 y 2 = − 42 2 x 2 + 4 y 2 = 102 \left\{ \begin{array} { l } 8 x ^ { 2 } - 2 y ^ { 2 } = - 42 \2 x ^ { 2 } + 4 y ^ { 2 } = 102\end{array} ight. { 8 x 2 − 2 y 2 = − 42 2 x 2 + 4 y 2 = 102 A) ( 1 , 5 ) , ( 1 , − 5 ) ( 1,5 ) , ( 1 , - 5 ) ( 1 , 5 ) , ( 1 , − 5 ) B) ( 1 , 5 ) , ( − 1 , − 5 ) ( 1,5 ) , ( - 1 , - 5 ) ( 1 , 5 ) , ( − 1 , − 5 ) C) ( − 1 , 5 ) , ( − 1 , − 5 ) ( - 1,5 ) , ( - 1 , - 5 ) ( − 1 , 5 ) , ( − 1 , − 5 ) D) ( 1 , 5 ) , ( 1 , − 5 ) , ( − 1 , 5 ) , ( − 1 , − 5 ) ( 1,5 ) , ( 1 , - 5 ) , ( - 1,5 ) , ( - 1 , - 5 ) ( 1 , 5 ) , ( 1 , − 5 ) , ( − 1 , 5 ) , ( − 1 , − 5 ) E) no solution exists

Question

Solve the system by the method of elimination.   { 8 x 2 − 2 y 2 = − 42 2 x 2 + 4 y 2 = 102 \left\{ \begin{array} { l } 8 x ^ { 2 } - 2 y ^ { 2 } = - 42 \2 x ^ { 2 } + 4 y ^ { 2 } = 102\end{array} ight. { 8 x 2 − 2 y 2 = − 42 2 x 2 + 4 y 2 = 102 ​ A)    ( 1 , 5 )  , ( 1 , − 5 )  ( 1,5 )  , ( 1 , - 5 )  ( 1 , 5 )  , ( 1 , − 5 )  B)    ( 1 , 5 )  , ( − 1 , − 5 )  ( 1,5 )  , ( - 1 , - 5 )  ( 1 , 5 )  , ( − 1 , − 5 )  C)    ( − 1 , 5 )  , ( − 1 , − 5 )  ( - 1,5 )  , ( - 1 , - 5 )  ( − 1 , 5 )  , ( − 1 , − 5 )  D)    ( 1 , 5 )  , ( 1 , − 5 )  , ( − 1 , 5 )  , ( − 1 , − 5 )  ( 1,5 )  , ( 1 , - 5 )  , ( - 1,5 )  , ( - 1 , - 5 )  ( 1 , 5 )  , ( 1 , − 5 )  , ( − 1 , 5 )  , ( − 1 , − 5 )  E) no solution exists

Gerard Wells · Accepted Answer

Final Answer : D, Explanation :   We can eliminate $y$ by multiplying the first equation by $2$ and the second equation by $-1$ and adding them together: ( 2 ) ( 8 x 2 − 2 y 2 ) + ( − 1 ) ( 2 x 2 + 4 y 2 ) = ( − 84 ) + ( − 102 ) 16 x 2 − 4 y 2 − 2 x 2 − 4 y 2 = − 186 14 x 2 = 2 x 2 = 1 7  (2)(8 x ^ { 2 } - 2 y ^ { 2 }) + (-1)(2 x ^ { 2 } + 4 y ^ { 2 })  & = (-84)+( -102)\16 x ^ { 2 } - 4 y ^ { 2 } - 2 x ^ { 2 } - 4 y ^ { 2 }  & = -186\14 x ^ { 2 }  & = 2\x ^ { 2 }  & = \frac{1}{7}  ( 2 ) ( 8 x 2 − 2 y 2 ) + ( − 1 ) ( 2 x 2 + 4 y 2 ) 16 x 2 − 4 y 2 − 2 x 2 − 4 y 2 14 x 2 x 2 ​ = ( − 84 ) + ( − 102 ) = − 186 = 2 = 7 1 ​ ​ Substituting this into the first equation, we have $8(\frac{1}{7})-2y^2=-42$, so $y^2=25$.Therefore, the solutions are $(x,y)=(\pm 1, \pm 5)$. Thus, the answer is D.

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