Question

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. ﻿{x=(y+2)2−1(x−2)2+(y+2)2=1
\begin{cases}
x = (y + $$2)^2 - 1$$ \
(x - $$2)^2 + (y$$ + $$2)^2 = 1$$
\end{cases}
{x=(y+2)2−1(x−2)2+(y+2)2=1​

Accepted Answer

Identify the two equations in the system: the first is $$x = (y + 2)^2$ - 1$$, which represents a parabola, and the second is $$(x - 2)^2$ + (y + 2)^2$ = 1$$, which represents a circle centered at $$(2, -2)$$ with radius 1. To find the points of intersection, substitute the expression for $$x$$ from the first equation into the second equation. This means replacing $$x in $$the circle's equation with $$(y + 2)^2$ - 1. $$After substitution, you will get an equation involving only $$y$$: $$(( (y + 2)^2$ - 1 ) - 2)^2$ + (y + 2)^2$ = 1. $$Simplify this equation step-by-step to form a polynomial in terms of $$y. $$Solve the resulting polynomial equation for $$y. $$Each real solution for $$y$$ corresponds to a point where the parabola and circle intersect. For each $$y$$ value found, substitute back into the first equation $$x = (y + 2)^2$ - 1 to $$find the corresponding $$x$$ coordinate. Check each $$(x, y)$$ pair by substituting them into both original equations to verify that they satisfy both equations, confirming the points of intersection and thus the solution set of the system.

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $\begin{cases}\)x = (y + 2)^2 - 1 \\(x - 2)^2 + (y + 2)^2 = 1\(\end{cases}$

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Key Concepts

Graphing Equations in the Coordinate Plane

Systems of Equations and Intersection Points

Checking Solutions by Substitution