Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
$x^2+y^2=5$
$-3x+4y=2$

Verified step by step guidance

1. Start with the given system of equations: \\ \(x^2 + y^2 = 5\) and \(-3x + 4y = 2\).
2. From the linear equation \(-3x + 4y = 2\), solve for one variable in terms of the other. For example, solve for \(y\): \\ \$4y = 2 + 3x\( \\ \)y = \frac{2 + 3x}{4}$.
3. Substitute the expression for \(y\) into the first equation \(x^2 + y^2 = 5\) to get an equation in terms of \(x\) only: \\ \(x^2 + \left(\frac{2 + 3x}{4}\right)^2 = 5\).
4. Simplify the equation by expanding the square and multiplying through by 16 (the denominator squared) to clear fractions, resulting in a quadratic equation in \(x\).
5. Solve the quadratic equation for \(x\) using the quadratic formula or factoring, then substitute each \(x\) value back into \(y = \frac{2 + 3x}{4}\) to find the corresponding \(y\) values. This will give all solutions, including complex ones if they exist.

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 beyond simple substitution or elimination used for linear systems, often involving substitution, factoring, or using the quadratic formula.

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, which can then be solved using algebraic techniques.

Complex Solutions

When solving nonlinear systems, solutions may include complex numbers if the equations yield negative values under square roots or other operations. Understanding how to work with complex numbers ensures all possible solutions, including nonreal ones, are found.