Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. y = x^2 - 2x + 1 x - 3y = -1

Key Concepts

Nonlinear Equations

Nonlinear equations are equations in which the variable(s) are raised to a power greater than one or involve products of variables. Unlike linear equations, which graph as straight lines, nonlinear equations can produce curves, parabolas, or other complex shapes. Understanding how to manipulate and solve these equations is crucial for finding their intersections or solutions.

Substitution Method

The substitution method is a technique used to solve systems of equations, where one equation is solved for one variable, and that expression is substituted into the other equation. This method simplifies the system, allowing for easier solving of nonlinear equations. It is particularly useful when one equation is already expressed in terms of one variable, as seen in the given problem.

Complex Solutions

Complex solutions arise when the solutions to an equation involve imaginary numbers, typically represented as 'a + bi', where 'a' and 'b' are real numbers and 'i' is the imaginary unit. In the context of nonlinear systems, it is important to recognize that some equations may yield solutions that are not real numbers, and understanding how to identify and express these complex solutions is essential for a complete solution set.