Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
$5x^2-2y^2=25$
$10x^2+y^2=50$

Verified step by step guidance

1. Start by writing down the system of equations clearly: \$5x^2 - 2y^2 = 25\( \)10x^2 + y^2 = 50$
2. To solve the system, consider using substitution or elimination. Notice both equations involve \(x^2\) and \(y^2\), so treat \(x^2\) and \(y^2\) as variables to simplify the process.
3. Multiply the first equation by a suitable number so that the coefficients of either \(x^2\) or \(y^2\) match in both equations. For example, multiply the first equation by 1 to keep it as is, and the second equation by 2 to align the \(y^2\) terms:
4. \$5x^2 - 2y^2 = 25\( \)20x^2 + 2y^2 = 100$
5. Add the two equations to eliminate \(y^2\): \((5x^2 - 2y^2) + (20x^2 + 2y^2) = 25 + 100\) Simplify to find an equation in terms of \(x^2\) only, then solve for \(x^2\). After finding \(x^2\), substitute back into one of the original equations to solve for \(y^2\). Finally, take square roots to find \(x\) and \(y\), remembering to consider both positive and negative roots as well as complex solutions.

Key Concepts

Nonlinear Systems of Equations

A nonlinear system consists of two or more equations involving variables raised to powers other than one or multiplied together. Solving such systems requires methods beyond simple substitution or elimination used for linear systems, often involving algebraic manipulation or substitution to reduce the system to solvable equations.

Substitution and Elimination Methods

These are algebraic techniques used to solve systems of equations. Substitution involves solving one equation for a variable and substituting into the other, while elimination involves adding or subtracting equations to eliminate a variable. Both methods help reduce the system to a single equation in one variable.

Complex Solutions in Algebra

When solving equations, solutions may include nonreal complex numbers, especially if the equation involves squares or other powers that can yield negative values under radicals. Recognizing and including complex solutions ensures a complete solution set, which is important in college algebra.