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Nonlinear Systems of Equations and ProblemSolving
As with linear systems, a nonlinear system of equations (and conics) can be solved graphically and algebraically for all its variables.
Learning Objective

Solve nonlinear systems of equations graphically and algebraically
Key Points

Subtracting one equation from another is an effective means for solving linear systems, but it often is difficult to use in nonlinear systems, in which the terms of two equations may be very different.

Substitution of a variable into another equation is usually the best method for solving nonlinear systems of equations.

Nonlinear systems of equations may have one or multiple solutions.
Terms

conic section
Any of the four distinct shapes that are the intersections of a cone with a plane, namely the circle, ellipse, parabola and hyperbola.

system of equations
A set of equations with multiple variables which can be solved using a specific set of values.
Full Text
A conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. There are a number of other geometric definitions possible. Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focusdirectrix definition of a conic, the circle is a limiting case with eccentricity 0. In modern geometry, certain degenerate cases, such as the union of two lines, are included as conics as well.
In a system of equations, two or more relationships are stated among variables. A system is solvable so long as there are as many simultaneous equations as variables. If each equation is graphed, the solution for the system can be found at the point where all the functions meet. The solution can be found either by inspection of a graph, typically with the use of software, or algebraically.
Nonlinear systems of equations, such as conic sections, include at least one function that is nonlinear. Because at least one function has curvature, it is possible for nonlinear systems of equations to contain multiple solutions. As with linear systems of equations, substitution can be used to solve nonlinear systems for one variable and then the other.
Solving nonlinear systems of equations algebraically is similar to doing the same for linear systems of equations. However, subtraction of one equation from another can become impractical if the two equations have different terms, which is more commonly the case in nonlinear systems.
Consider, for example, the following system of equations :
Substituting x^{2} for y in equation 2:
This quadratic equation can be solved by moving all the equation's components to the left before using the quadratic formula:
Using the quadratic formula, with a=1, b=2 and c=6, it can be determined that x=3 and x=2 are solutions.
The solutions for x can then be plugged into either of the original systems to find the value of y. In this example, we will use equation 1:
Thus, for x=2, y=4. And for x=3, y=9.
Our final solutions are: (2, 4) and (3, 9).
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Key Term Reference
 Interest
 Appears in this related concepts: Interest Compounded Continuously, Debt Utilization Ratios, and Tax Considerations
 circle
 Appears in this related concepts: Circles, Applications and ProblemSolving, and Circles
 conical
 Appears in this related concepts: Quadratic Functions of the Form f(x) = ax^2 + bx + c, Where a is not Equal to 0, Parabolas, and Applications and ProblemSolving
 degree
 Appears in this related concepts: Polynomials: Introduction, Addition, and Subtraction, Solving Quadratic Equations by Factoring, and Partial Fractions
 ellipse
 Appears in this related concepts: Conic Sections, Planetary Motion According to Kepler and Newton, and Ellipses
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 function
 Appears in this related concepts: Solving Differential Equations, Four Ways to Represent a Function, and Average Value of a Function
 geometric
 Appears in this related concepts: Addition, Subtraction, and Multiplication, Transformations of Functions, and Sums and Series
 graph
 Appears in this related concepts: Graphing on Computers and Calculators, Reading Points on a Graph, and Graphing Equations
 hyperbola
 Appears in this related concepts: Conic Sections in Polar Coordinates, Inverse Variation, and Applications and ProblemSolving
 linear
 Appears in this related concepts: Exponential Growth and Decay, Linear Approximation, and Delivery Tips
 linear system
 Appears in this related concepts: Solving Systems of Equations in Three Variables, Nonlinear Systems of Inequalities, and Inconsistent and Dependent Systems
 parabola
 Appears in this related concepts: Standard Form and Completing the Square, The Quadratic Formula, and Standard Equations of Hyperbolas
 point
 Appears in this related concepts: The Intermediate Value Theorem, Introduction: Polynomial and Rational Functions and Models, and Quadratic Functions of the Form f(x) = a(xh)^2 + k
 quadratic
 Appears in this related concepts: Stretching and Shrinking, The Discriminant, and Quadratic Equations and Quadratic Functions
 quadratic equation
 Appears in this related concepts: Zeroes of Polynomial Functions with Real Coefficients, Completing the Square, and Linear and Quadratic Equations
 term
 Appears in this related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Democracy
 variable
 Appears in this related concepts: Related Rates, Calculating the NPV, and Controlling for a Variable
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Source: Boundless. “Nonlinear Systems of Equations and ProblemSolving.” Boundless Algebra. Boundless, 27 Jun. 2014. Retrieved 28 Apr. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/conicsections7/nonlinearsystemsofequationsandinequalities52/nonlinearsystemsofequationsandproblemsolving22111097/