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.﻿{x2+y2=1x2+9y2=9\begin{cases}
$$x^2$$ + $$y^2 = 1$$ \
$$x^2$$ + $$9y^2 = 9$$
\end{cases}{x2+y2=1x2+9y2=9​

Accepted Answer

Identify the two equations in the system: the first is $$x^{2}$$ + $$y^{2} = 1$, which represents a circle centered at the origin with radius 1, and the second is x$$^{2}$$ + 9y$$^{2}$$ = 9$$, which represents an ellipse centered at the origin. To find the points of intersection, set up the system of equations and consider subtracting one equation from the other to eliminate $$x^{2} or$$ $$y^{2}. $$For example, subtract the first equation from the second: $$\$$left(x^{2}$$ + 9y$$^{2}$$\right) - \$$left(x^{2}$$ + y$$^{2}$$\right) = 9 - 1. $$Simplify the subtraction to get an equation involving only $$y^{2}$$: $$9y^{2}$$ - $$y^{2} = 8$, which simplifies to 8y$$^{2}$$ = 8. $$Solve for $$y^{2} to $$find $$y^{2} = 1$. Use the values of y$$^{2}$$ found to substitute back into one of the original equations (for example, x$$^{2}$$ + y$$^{2}$$ = 1) to $$solve for $$x^{2}. $$This will give you the corresponding $$x$$ values for each $$y. $$List all possible $$(x, y)$$ pairs from the previous step as the intersection points. Finally, check each point by substituting back into both original equations to verify they satisfy both equations.

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^2 + y^2 = 1 \(\x\)^2 + 9y^2 = 9\(\end{cases}$

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Key Concepts

Graphing Conic Sections

Systems of Equations

Checking Solutions

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.{x2+y2=1x2+9y2=9\(\begin{cases}\)x^2 + y^2 = 1 \(\x\)^2 + 9y^2 = 9\(\end{cases}\){x2+y2=1x2+9y2=9