Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
$x^{2} + y^{2} = 102$
$x^{2} - y^{2} = 17$

Verified step by step guidance

Start by writing down the system of equations: \[x^2 + y^2 = 10\] \[2x^2 - y^2 = 17\]

Add the two equations together to eliminate $y^2$. Adding gives: \[ (x^2 + y^2) + (2x^2 - y^2) = 10 + 17 \] which simplifies to: \[ 3x^2 = 27 \]

Solve for $x^2$ by dividing both sides by 3: \[ x^2 = 9 \]

Find the values of $x$ by taking the square root of both sides: \[ x = \pm 3 \]

Substitute each value of $x$ back into one of the original equations (for example, $x^2 + y^2 = 10$) to solve for $y^2$, then find $y$ by taking the square root, remembering to consider both positive and negative roots.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Nonlinear Systems of Equations

A nonlinear system involves equations where variables are 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 forms.

Substitution and Elimination Methods

These are techniques to solve systems of equations by expressing one variable in terms of another (substitution) or combining equations to eliminate a variable (elimination). For nonlinear systems, these methods help transform the system into a single equation in one variable, simplifying the solution process.

Complex Number Solutions

Nonlinear systems can have solutions that are not real numbers but complex numbers involving the imaginary unit i. Recognizing when to include complex solutions is essential, especially when the system’s equations lead to negative values under square roots or other operations that extend beyond the real number set.

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