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

Verified step by step guidance

1. 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.
2. 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: \(\left(x^{2} + 9y^{2}\right) - \left(x^{2} + y^{2}\right) = 9 - 1\).
3. 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$.
4. 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 \(y\).
5. 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.

Key Concepts

Graphing Conic Sections

Graphing conic sections involves plotting curves defined by quadratic equations in two variables. In this problem, the equations represent a circle and an ellipse, which are types of conic sections. Understanding their standard forms and shapes helps in sketching accurate graphs to find intersection points.

Systems of Equations

A system of equations consists of two or more equations with the same variables. The solution set includes all points that satisfy every equation simultaneously. Graphically, solutions correspond to points where the graphs intersect, making it essential to identify these points accurately.

Checking Solutions

After finding potential solutions from the graph, substituting these points back into the original equations verifies their validity. This step ensures that the intersection points truly satisfy both equations, confirming the solution set for the system.