Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
$3 x^{2} + 5 y^{2} = 17$
$2 x^{2} - 3 y^{2} = 5$

검증된 단계별 안내

Start by writing down the system of equations clearly: \[3x^2 + 5y^2 = 17\] \[2x^2 - 3y^2 = 5\]

Introduce new variables to simplify the system: let \[a = x^2\] and \[b = y^2\]. This transforms the system into: \[3a + 5b = 17\] \[2a - 3b = 5\]

Solve the system of linear equations for \[a\] and \[b\] using either substitution or elimination. For example, multiply the second equation by 5 and the first by 3 to align coefficients of \[b\], then add or subtract to eliminate \[b\].

Once you find the values of \[a\] and \[b\], recall that \[a = x^2\] and \[b = y^2\]. Solve for \[x\] and \[y\] by taking the square roots of \[a\] and \[b\] respectively, remembering to consider both positive and negative roots as well as complex roots if \[a\] or \[b\] are negative.

Write down all possible solutions for \[x\] and \[y\], including real and nonreal complex solutions, based on the values obtained for \[a\] and \[b\].

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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 Number Solutions

When solving nonlinear systems, solutions may include complex numbers if the equations yield negative values under square roots or other operations. Complex solutions have a real part and an imaginary part, and including them ensures all possible solutions to the system are found.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. 3x2+5y2=173x^2 + 5y^2 = 172x2−3y2=52x^2 - 3y^2 = 5