Solve the nonlinear system of equations.
-$$\left\{ \begin{array} { l }
10 x ^ { 2 } - 5 y ^ { 2 } = 15 \
- x ^ { 2 } + y ^ { 2 } = 1
\end{array} \right.$$
A) $( 2 , \sqrt { 5 } ) , ( 2 , - \sqrt { 5 } ) , ( - 2 , \sqrt { 5 } ) , ( - 2 , - \sqrt { 5 } )$
B) $( \sqrt { 5 } , 2 ) , ( \sqrt { 5 } , - 2 ) , ( - \sqrt { 5 } , 2 ) , ( - \sqrt { 5 } , - 2 )$
C) $( 2 , \sqrt { 5 } ) , ( - 2 , - \sqrt { 5 } )$
D) $( 2 , \sqrt { 5 } ) , ( 2 , - \sqrt { 5 } )$

Question

Solve the nonlinear system of equations.
-$$\left\{ \begin{array} { l } 
10 x ^ { 2 } - 5 y ^ { 2 } = 15 \
- x ^ { 2 } + y ^ { 2 } = 1
\end{array} \right.$$
A) $( 2 , \sqrt { 5 } ) , ( 2 , - \sqrt { 5 } ) , ( - 2 , \sqrt { 5 } ) , ( - 2 , - \sqrt { 5 } )$
B) $( \sqrt { 5 } , 2 ) , ( \sqrt { 5 } , - 2 ) , ( - \sqrt { 5 } , 2 ) , ( - \sqrt { 5 } , - 2 )$
C) $( 2 , \sqrt { 5 } ) , ( - 2 , - \sqrt { 5 } )$
D) $( 2 , \sqrt { 5 } ) , ( 2 , - \sqrt { 5 } )$

Quizplus · Accepted Answer

The system can be solved by expressing one equation in terms of one variable and substituting into the other. From the second equation, $y^2 = x^2 + 1$. Substituting this into the first equation gives $10x^2 - 5(x^2 + 1) = 15$, simplifying to $5x^2 = 20$, so $x^2 = 4$, and thus $x = \pm 2$. Substituting $x = \pm 2$ back into $y^2 = x^2 + 1$ gives $y^2 = 5$, so $y = \pm \sqrt{5}$. This results in four possible combinations of $x$ and $y$: $(2, \sqrt{5})$, $(2, -\sqrt{5})$, $(-2, \sqrt{5})$, and $(-2, -\sqrt{5})$.

Solve the Nonlinear System of Equations {10x2−5y2=15−x2+y2=1\left\{ \begin{array} { l } 10 x ^ { 2 } - 5 y ^ { 2 } = 15 \\- x ^ { 2 } + y ^ { 2 } = 1\end{array} \right.{10x2−5y2=15−x2+y2=1​

Solve the Nonlinear System of Equations $\left\{ \begin{array} { l } 10 x ^ { 2 } - 5 y ^ { 2 } = 15 \\- x ^ { 2 } + y ^ { 2 } = 1\end{array} \right.$