Answer
$(0,-1), \left(\dfrac{5}{2},-\dfrac{7}{2}\right)$
Work Step by Step
We are given the system:
$\begin{cases}
x+y+1=0\\
x^2+y^2+6y-x=-5
\end{cases}$
We will use the substitution method. Solve Equation 1 for $x$. Substitute the expression of $x$ in Equation 2 to eliminate $x$ and determine $y$:
$\begin{cases}
x=-y-1\\
(-y-1)^2+y^2+6y-(-y-1)=-5
\end{cases}$
$y^2+2y+1+y^2+6y+y+1=-5$
$2y^2+9y+7=0$
$2y^2+2y+7y+7=0$
$2y(y+1)+7(y+1)=0$
$(y+1)(2y+7)=0$
$y+1=0\Rightarrow y_1=-1$
$2y+7=0\Rightarrow y_2=-\dfrac{7}{2}$
Substitute each of the values of $y$ in the expression of $x$ to determine $x$:
$x=-y-1$
$y_1=-1\Rightarrow x_1=-(-1)-1=0$
$y_2=-\dfrac{7}{2}\Rightarrow x_2=-\left(-\dfrac{7}{2}\right)-1=\dfrac{5}{2}$
The system's solutions are:
$(0,-1), \left(\dfrac{5}{2},-\dfrac{7}{2}\right)$
