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 of 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

system of equations
A set of formulas with multiple variables which can be solved using a specific set of values.

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.

nonlinear
An algebraic term that is raised to the power of two or higher; equivalently, a function with a curved graph.
Full Text
Conic Sections
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. The four types of conic section are the hyperbola, the parabola, the ellipse, and the circle, though the circle can be considered to be a special case of the ellipse.
The type of a conic corresponds to its eccentricity. Conics with eccentricity less than
System of Equations
In a system of equations, two or more relationships are stated among variables. A system is solvable as 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 by using graphing or plotting software, or algebraically.
Nonlinear Systems
Nonlinear systems of equations, such as conic sections, include at least one equation that is nonlinear. A nonlinear equation is defined as an equation possessing at least one term that is raised to a power of 2 or more. When graphed, these equations produce curved lines.
Since 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.
Example
Consider, for example, the following system of equations:
Integer values of $y=x^2$ (blue) and $y=x+6$ (red)
The parabola (blue) falls below the line (red) between
We can solve this system algebraically by using equation
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
The solutions for
Thus, for
Our final solutions are:
Key Term Reference
 Parabola
 Appears in these related concepts: Restricting Domains to Find Inverses, Parts of an Ellipse, and Conics in Polar Coordinates
 circle
 Appears in these related concepts: Introduction to Circles, Types of Conic Sections, and Applications of Circles and Ellipses
 conical
 Appears in these related concepts: Parabolas As Conic Sections and Applications of the Parabola
 degenerate
 Appears in these related concepts: Crystal Field Theory, Wave Equation for the Hydrogen Atom, and The Central Dogma: DNA Encodes RNA and RNA Encodes Protein
 degree
 Appears in these related concepts: Solving Quadratic Equations by Factoring, Partial Fractions, and Adding and Subtracting Polynomials
 eccentricity
 Appears in these related concepts: Conic Sections, Kepler's First Law, and Satellites
 ellipse
 Appears in these related concepts: Planetary Motion According to Kepler and Newton, Ellipses, and Conic Sections in Polar Coordinates
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Graphs of Equations as Graphs of Solutions
 function
 Appears in these related concepts: Visualizing Domain and Range, The Vertical Line Test, and Solving Differential Equations
 geometric
 Appears in these related concepts: Addition, Subtraction, and Multiplication, Sums and Series, and Introduction to Sequences
 graph
 Appears in these related concepts: Graphical Representations of Functions, Graphing Equations, and Reading Points on a Graph
 hyperbola
 Appears in these related concepts: Applications of Hyperbolas, Direct and Inverse Variation, and Introduction to Hyperbolas
 linear
 Appears in these related concepts: Exponential Growth and Decay, Graphs of Linear Inequalities, and Factoring General Quadratics
 linear system
 Appears in these related concepts: Solving Systems of Equations in Three Variables, Nonlinear Systems of Inequalities, and Inconsistent and Dependent Systems
 parabola
 Appears in these related concepts: Parts of a Parabola, The Quadratic Formula, and Completing the Square
 point
 Appears in these related concepts: Graphing Quadratic Equations In Standard Form, The Intermediate Value Theorem, and Polynomial and Rational Functions as Models
 quadratic
 Appears in these related concepts: The Discriminant, Stretching and Shrinking, and What is a Quadratic Function?
 quadratic equation
 Appears in these related concepts: Linear and Quadratic Equations, Zeroes of Polynomial Functions with Real Coefficients, and Standard Form and Completing the Square
 solution
 Appears in these related concepts: Electrolyte and Nonelectrolyte Solutions, Turning Your Claim Into a Thesis Statement, and What is an Equation?
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Introduction to Variables
 variable
 Appears in these related concepts: What is a Linear Function?, Fundamentals of Statistics, and Math Review
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