\( x ^ { 2 } + y ^ { 2 } - 1 \)

This equation represents a circle with a radius of 1 centered at the origin (0,0) in the Cartesian plane. It’s a classic example used in algebra and geometry to illustrate the relationship between the coordinates \(x\) and \(y\). If you plot it out, you’ll see that every point on the circle satisfies this equation! To dive deeper, consider how this circle intersects with linear equations, forming a fascinating partnership in determining the properties of the resulting shapes. For instance, when lines intersect the circle, you can explore solutions that yield unique points or even no solutions at all, demonstrating key concepts of geometry and algebraic relationships!