Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components.
2xy + 1 = 0
x + 16y = 2

Verified step by step guidance

1. Start with the given system of equations: \[2xy + 1 = 0\] \[x + 16y = 2\]
2. From the second equation, solve for one variable in terms of the other. For example, solve for \[x\]: \[x = 2 - 16y\]
3. Substitute the expression for \[x\] into the first equation to eliminate \[x\]: \[2(2 - 16y)y + 1 = 0\]
4. Simplify the resulting equation to get a quadratic equation in terms of \[y\]: \[2(2y - 16y^2) + 1 = 0\] which simplifies to \[4y - 32y^2 + 1 = 0\]
5. Solve the quadratic equation for \[y\] using the quadratic formula: \[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] where \[a = -32\], \[b = 4\], and \[c = 1\]. Then substitute each \[y\] value back into \[x = 2 - 16y\] to find the corresponding \[x\] values.

Key Concepts

Nonlinear Systems of Equations

A nonlinear system involves at least one equation that is not linear, meaning variables may be multiplied together or raised to powers other than one. Solving such systems requires methods beyond simple substitution or elimination used for linear systems, often involving algebraic manipulation or substitution to reduce to a solvable form.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can be solved using algebraic techniques, including handling nonlinear terms.

Complex Solutions

When solving nonlinear systems, solutions may include complex numbers, especially if the equations lead to quadratic or higher-degree polynomials with no real roots. Understanding how to work with complex numbers, including imaginary units, is essential to find all possible solutions.