Solve the system of equations: \(x + y + 2 = 0 , x ^ { 2 } + y ^ { 2 } + 4 y - 3 x = - 4\) :\(\left ( \begin{array} { l l } { x = \frac { 3 } { 2 } , } & { y = - \frac { 7 } { 2 } } \\ { x = 0 , } & { y = - 2 } \end{array} \right ) \)
Steps
\(\left [ \begin{array} { c } { x + y + 2 = 0 } \\ { x ^ { 2 } + y ^ { 2 } + 4 y - 3 x = - 4 } \end{array} \right ] \)
Isolate \(y\) for \(x + y + 2 = 0 : y = - x - 2\)
Plug the solutions \(y = - x - 2\) into \(x ^ { 2 } + y ^ { 2 } + 4 y - 3 x = - 4\)
For \(x ^ { 2 } + y ^ { 2 } + 4 y - 3 x = - 4\) , subsitute \(y\) with \(- x - 2 : x = \frac { 3 } { 2 } , x = 0\)
Plug the solutions \(x = \frac { 3 } { 2 } , x = 0\) into \(x + y + 2 = 0\)
For \(x + y + 2 = 0\) , subsitute \(x\) with \(\frac { 3 } { 2 } : y = - \frac { 7 } { 2 } \)
For \(x + y + 2 = 0\) , subsitute \(x\) with \(0 : y = - 2\)
Verify solutions by plugging them into the original equations
Therefore, the final solutions for \(x + y + 2 = 0 , x ^ { 2 } + y ^ { 2 } + 4 y - 3 x = - 4\) are
\(\left ( \begin{array} { l } { x = \frac { 3 } { 2 } , y = - \frac { 7 } { 2 } } \\ { x = 0 , y = - 2 } \end{array} \right ) \)