Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components.
y = x² - 2x + 1
x - 3y = -1

Verified step by step guidance

1. Start with the given system of equations: \(y = x^{2} - 2x + 1\) and \(x - 3y = -1\).
2. Substitute the expression for \(y\) from the first equation into the second equation to eliminate \(y\). This gives: \(x - 3(x^{2} - 2x + 1) = -1\).
3. Expand and simplify the equation: \(x - 3x^{2} + 6x - 3 = -1\) which simplifies to \(-3x^{2} + 7x - 3 = -1\).
4. Bring all terms to one side to form a quadratic equation: \(-3x^{2} + 7x - 3 + 1 = 0\) which simplifies to \(-3x^{2} + 7x - 2 = 0\).
5. Solve the quadratic equation for \(x\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) where \(a = -3\), \(b = 7\), and \(c = -2\). After finding the values of \(x\), substitute each back into the original equation for \(y\) to find the corresponding \(y\) values.

Key Concepts

Nonlinear Systems of Equations

A nonlinear system involves at least one equation that is not linear, such as quadratic or higher-degree polynomials. Solving these systems requires methods that handle curves and more complex relationships, unlike linear systems which involve straight lines.

Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve, especially when one equation is already solved for a variable.

Complex Solutions

When solving polynomial equations, solutions may include nonreal complex numbers involving the imaginary unit i. Recognizing and including complex solutions ensures all possible roots are found, which is essential for a complete solution to nonlinear systems.