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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 :
Integer values of y=x^2 (blue) and y=x+6 (red)
The parabola (blue) falls below the line (red) between x=2 and x=3. For all values of x less than 2 and greater than 3, the parabola is greater than the line.
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 these related concepts: Accounting for Interest Earned and Principal at Maturity, Interest Compounded Continuously, and Tax Considerations
 circle
 Appears in these related concepts: Circles, Applications and ProblemSolving, and Circles
 conical
 Appears in these 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 these related concepts: Experimental Probabilities, Polynomials: Introduction, Addition, and Subtraction, and Partial Fractions
 ellipse
 Appears in these related concepts: Conic Sections, Planetary Motion According to Kepler and Newton, and Ellipses
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Functions and Their Notation
 geometric
 Appears in these related concepts: Addition, Subtraction, and Multiplication, Transformations of Functions, and Sums and Series
 graph
 Appears in these related concepts: Graphing on Computers and Calculators, Reading Points on a Graph, and Graphing Functions
 hyperbola
 Appears in these related concepts: Conic Sections in Polar Coordinates, Standard Equations of Hyperbolas, and Applications and ProblemSolving
 linear
 Appears in these related concepts: Trinomials of the Form ax^2 + bx + c, Where a is Not Equal to 1, Exponential Growth and Decay, and Graphs of Linear Inequalities
 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: Standard Form and Completing the Square, Completing the Square, and The Quadratic Formula
 point
 Appears in these related concepts: The Intermediate Value Theorem, Relative Minima and Maxima, and Graphing Equations
 quadratic
 Appears in these related concepts: Solving Quadratic Equations by Factoring, The Discriminant, and Quadratic Functions of the Form f(x) = a(xh)^2 + k
 quadratic equation
 Appears in these related concepts: Zeroes of Polynomial Functions with Real Coefficients, Solving Equations with Rational Expressions; Problems Involving Proportions, and Linear and Quadratic Equations
 term
 Appears in these related concepts: Arithmetic Sequences, Basics of Graphing Polynomial Functions, and The 22nd Amendment
 variable
 Appears in these related concepts: Related Rates, Math Review, and Psychology and the Scientific Method: From Theory to Conclusion
Sources
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Cite This Source
Source: Boundless. “Nonlinear Systems of Equations and ProblemSolving.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 14 Oct. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/conicsections7/nonlinearsystemsofequationsandinequalities52/nonlinearsystemsofequationsandproblemsolving22111097/