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

Verified step by step guidance

1. <Step 1: Identify the system of equations.> $3x^2+5y^2=17$ and $2x^2-3y^2=5$.
2. <Step 2: Use the elimination method to eliminate one of the variables.> Multiply the first equation by 2 and the second equation by 3 to align the coefficients of $x^2$.
3. <Step 3: Subtract the modified second equation from the modified first equation to eliminate $x^2$.> This will give you an equation in terms of $y^2$ only.
4. <Step 4: Solve the resulting equation for $y^2$.> This will give you the possible values for $y^2$.
5. <Step 5: Substitute the values of $y^2$ back into one of the original equations to solve for $x^2$.> This will give you the possible values for $x^2$.

Key Concepts

Nonlinear Systems of Equations

A nonlinear system of equations consists of two or more equations where at least one equation is not linear, meaning it cannot be expressed in the form y = mx + b. These systems can involve polynomial, exponential, or trigonometric functions. Solving such systems often requires methods like substitution, elimination, or graphical analysis to find points of intersection that satisfy all equations.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part multiplied by the imaginary unit 'i', which is defined as the square root of -1. In the context of solving equations, complex solutions arise when the discriminant of a quadratic equation is negative, indicating that the solutions cannot be represented on the real number line.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically written in the standard form ax^2 + bx + c = 0. They can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. In the context of the given system, the equations can be manipulated to form quadratic equations, allowing for the identification of both real and complex solutions.