Implicit equations are mathematical expressions where the relationship between variables is defined without explicitly solving for one variable in terms of another. For example, in the equation x² + y² = 1, both x and y are interdependent, and the equation describes a circle without isolating either variable. These equations are often used in geometry and calculus to represent curves and surfaces. They can describe complex shapes and relationships that are not easily expressed in explicit form. Understanding implicit equations is essential for fields like physics, engineering, and computer graphics, where such relationships frequently arise.