Solve Nonlinear Systems By Addition
-$$\begin{array} { l }
2 x ^ { 2 } + x y - y ^ { 2 } = 3 \
x ^ { 2 } + 2 x y + y ^ { 2 } = 3
\end{array}$$
A) $\left\{ \left( \frac { 2 \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } \right) , \left( - \frac { 2 \sqrt { 3 } } { 3 } , - \frac { \sqrt { 3 } } { 3 } \right) \right\}$
B) $\left\{ \left( \frac { 2 \sqrt { 3 } } { 3 } , - \frac { \sqrt { 3 } } { 3 } \right) , \left( - \frac { 2 \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } \right) \right\}$
C) $\left\{ \left( \frac { - 2 } { 3 } , \frac { 1 } { 3 } \right) , \left( \frac { 2 \sqrt { 3 } } { 3 } , - \frac { 1 } { 3 } \right) \right\}$
D) $\left\{ \left( \frac { - 2 } { 3 } , - \frac { 1 } { 3 } \right) , \left( \frac { 2 \sqrt { 3 } } { 3 } , \frac { 1 } { 3 } \right) \right\}$

Question

Solve Nonlinear Systems By Addition
-$$\begin{array} { l } 
2 x ^ { 2 } + x y - y ^ { 2 } = 3 \
x ^ { 2 } + 2 x y + y ^ { 2 } = 3
\end{array}$$
A) $\left\{ \left( \frac { 2 \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } \right) , \left( - \frac { 2 \sqrt { 3 } } { 3 } , - \frac { \sqrt { 3 } } { 3 } \right) \right\}$
B) $\left\{ \left( \frac { 2 \sqrt { 3 } } { 3 } , - \frac { \sqrt { 3 } } { 3 } \right) , \left( - \frac { 2 \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } \right) \right\}$
C) $\left\{ \left( \frac { - 2 } { 3 } , \frac { 1 } { 3 } \right) , \left( \frac { 2 \sqrt { 3 } } { 3 } , - \frac { 1 } { 3 } \right) \right\}$
D) $\left\{ \left( \frac { - 2 } { 3 } , - \frac { 1 } { 3 } \right) , \left( \frac { 2 \sqrt { 3 } } { 3 } , \frac { 1 } { 3 } \right) \right\}$

Quizplus · Accepted Answer

Adding the two equations directly doesn't simplify to a linear equation due to the nature of the terms involved. Instead, we can notice that the second equation is the expansion of $(x + y)^2$. This means we have a system where the first equation involves a difference of squares and the second a perfect square, which doesn't directly lead to a simple addition solution. However, solving the system using substitution or another method reveals that the solutions involve $\sqrt{3}$ components, matching option A.

Solve Nonlinear Systems by Addition - 2x2+xy−y2=3x2+2xy+y2=3\begin{array} { l } 2 x ^ { 2 } + x y - y ^ { 2 } = 3 \\x ^ { 2 } + 2 x y + y ^ { 2 } = 3\end{array}2x2+xy−y2=3x2+2xy+y2=3​

Solve Nonlinear Systems by Addition
- $\begin{array} { l } 2 x ^ { 2 } + x y - y ^ { 2 } = 3 \\x ^ { 2 } + 2 x y + y ^ { 2 } = 3\end{array}$