Solve the system of nonlinear equations by the addition method. $$\begin{array} { l }
x ^ { 2 } + y ^ { 2 } = 20 \
3 y ^ { 2 } - x = 50
\end{array}$$
A) $\left\{ ( - 2 , - 4 ) , ( - 2,4 ) , \left( \frac { 5 } { 3 } , - \frac { \sqrt { 155 } } { 3 } \right) , \left( \frac { 5 } { 3 } , \frac { \sqrt { 155 } } { 3 } \right) \right\}$
B) $\left\{ ( \sqrt { 20 } , 0 ) , \left( 0 , - \sqrt { \frac { 50 } { 3 } } \right) \right\}$
C) $\{ ( 0 , \sqrt { 20 } ) , ( - 2,4 ) \}$
D) { }

Question

Solve the system of nonlinear equations by the addition method. $$\begin{array} { l } 
x ^ { 2 } + y ^ { 2 } = 20 \
3 y ^ { 2 } - x = 50
\end{array}$$
A) $\left\{ ( - 2 , - 4 ) , ( - 2,4 ) , \left( \frac { 5 } { 3 } , - \frac { \sqrt { 155 } } { 3 } \right) , \left( \frac { 5 } { 3 } , \frac { \sqrt { 155 } } { 3 } \right) \right\}$
B) $\left\{ ( \sqrt { 20 } , 0 ) , \left( 0 , - \sqrt { \frac { 50 } { 3 } } \right) \right\}$
C) $\{ ( 0 , \sqrt { 20 } ) , ( - 2,4 ) \}$
D) { }

Quizplus · Accepted Answer

To solve the system, we can manipulate the equations to facilitate elimination or substitution. From the second equation, express $x$ in terms of $y^2$: $x = 3y^2 - 50$. Substitute this expression for $x$ in the first equation: $(3y^2 - 50)^2 + y^2 = 20$. This substitution leads to a quartic equation in terms of $y$, which can be solved to find the values of $y$, and subsequently, the corresponding $x$ values can be determined. The correct solutions include both real and complex pairs, indicating multiple intersection points between the circle defined by the first equation and the curve defined by the second equation.

Solve the System of Nonlinear Equations by the Addition Method x2+y2=203y2−x=50\begin{array} { l } x ^ { 2 } + y ^ { 2 } = 20 \\3 y ^ { 2 } - x = 50\end{array}x2+y2=203y2−x=50​

Solve the System of Nonlinear Equations by the Addition Method $\begin{array} { l } x ^ { 2 } + y ^ { 2 } = 20 \\3 y ^ { 2 } - x = 50\end{array}$