Determine the real solutions to the system of nonlinear equations.
-$$\begin{array} { l }
x ^ { 2 } - y ^ { 2 } = 27 \
x ^ { 2 } + y ^ { 2 } = 51
\end{array}$$
A) $( 0,0 )$
B) $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , 2 \sqrt { 3 } )$
C) $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , 2 \sqrt { 3 } ) , ( \sqrt { 39 } , - 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , - 2 \sqrt { 3 } )$
D) $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , - 2 \sqrt { 3 } )$

Question

Determine the real solutions to the system of nonlinear equations.
-$$\begin{array} { l } 
x ^ { 2 } - y ^ { 2 } = 27 \
x ^ { 2 } + y ^ { 2 } = 51
\end{array}$$
A) $( 0,0 )$
B) $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , 2 \sqrt { 3 } )$
C) $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , 2 \sqrt { 3 } ) , ( \sqrt { 39 } , - 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , - 2 \sqrt { 3 } )$
D) $( \sqrt { 39 } , 2 \sqrt { 3 } ) , ( - \sqrt { 39 } , - 2 \sqrt { 3 } )$

Quizplus · Accepted Answer

Adding the two equations gives $2x^2 = 78$, leading to $x^2 = 39$, so $x = \pm \sqrt{39}$. Substituting $x^2 = 39$ into either equation allows solving for $y^2$, yielding $y = \pm 2\sqrt{3}$. This results in four combinations of $x$ and $y$.

Determine the Real Solutions to the System of Nonlinear Equations x2−y2=27x2+y2=51\begin{array} { l } x ^ { 2 } - y ^ { 2 } = 27 \\x ^ { 2 } + y ^ { 2 } = 51\end{array}x2−y2=27x2+y2=51​

Determine the Real Solutions to the System of Nonlinear Equations $\begin{array} { l } x ^ { 2 } - y ^ { 2 } = 27 \\x ^ { 2 } + y ^ { 2 } = 51\end{array}$