Question

Graph each equation of the system. Then solve the system to find the points of intersection.

$\left\{\begin{array}{l}x+y+1=0\\ x^2+y^2+6y-x=-5\end{array}\right.$

Short answer

The point of intersection is $\left(0,-1\right)\&\left(\frac{5}{2},-\frac{7}{2}\right)$.

Step-by-step solution

Step 1. Given information

$\left\{\begin{array}{l}x+y+1=0\\ x^2+y^2+6y-x=-5\end{array}\right.$

Step 2. Graphing each equation

$x+y+1=0$

$x^2+y^2+6y-x=-5$

Step 3. Calculation of intersection point

$$\begin{gathered} Consider, \\ x+y+1=0.....\left(1\right) \\ {x}^2+y^2+6y-x=-5.....\left(2\right) \\ Nowsubstitutingthevalueofyfromeq.1intoeq.2 \\ weget \\ {x}^2+{(-x-1)}^2+6(-x-1)-x=-5 \\ {x}^2+x^2+1+2x-6x-6-x=-5 \\ x=0,\frac{5}{2} \\ Nowsubstitutingvalueofxineq.1weget \\ y=-1 \\ y=-\frac{7}{2} \end{gathered}$$