# Nonlinear Systems of Equations and Problem-Solving

## As with linear systems, a nonlinear system of equations (and conics) can be solved graphically and algebraically for all its variables.

#### 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

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

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

#### Figures

1. ##### 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.

2. ##### Interactive Graph

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 focus-directrix 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 non-linear. 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 (Figure 1):

$y=x^2$                                                                                                         (1)

$y=x+6$                                                                                                 (2)

Substituting x2 for y in equation 2:

$x^2=x+6$

This quadratic equation can be solved by moving all the equation's components to the left before using the quadratic formula:

$x^2-x-6=0$

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:

$y=(-2)^2$

$y=3^2$

Thus, for x=-2, y=4. And for x=3, y=9.

Our final solutions are: (-2, 4) and (3, 9).

#### Key Term Glossary

algebraic
or function}} Containing only numbers, letters and arithmetic operators.
##### Appears in these related concepts:
circle
A two-dimensional geometric figure, a line, consisting of the set of all those points in a plane that are equally distant from another point.
##### Appears in these related concepts:
conical
Shaped like a cone; of or relating to a cone or cones.
##### Appears in these related concepts:
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.
##### Appears in these related concepts:
degree
the sum of the exponents of a term; the order of a polynomial.
##### Appears in these related concepts:
directrix
A line used to define a curve or surface; especially a line, the distance from which a point on a conic has a constant ratio to that from the focus
##### Appears in these related concepts:
ellipse
A closed curve, the locus of a point such that the sum of the distances from that point to two other fixed points (called the foci of the ellipse) is constant; equivalently, the conic section that is the intersection of a cone with a plane that does not intersect the base of the cone.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equals sign. E.g. x=5.
##### Appears in these related concepts:
function
A relation in which each element of the input is associated with exactly one element of the output.
##### Appears in these related concepts:
geometric
increasing or decreasing in a geometric progression, i.e. multiplication by a constant.
##### Appears in these related concepts:
graph
A diagram displaying data; in particular one showing the relationship between two or more quantities, measurements or numbers.
##### Appears in these related concepts:
hyperbola
A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone.
##### Appears in these related concepts:
interest
The price paid for obtaining, or price received for providing, money or goods in a credit transaction, calculated as a fraction of the amount or value of what was borrowed.
##### Appears in these related concepts:
linear
Of or relating to a class of polynomial of the form y = ax + b .
##### Appears in these related concepts:
linear system
A mathematical model of a system based on the use of a linear operator.
##### Appears in these related concepts:
parabola
The conic section formed by the intersection of a cone with a plane parallel to a plane tangent to the cone; the locus of points equidistant from a fixed point (the focus) and line (the directrix).
##### Appears in these related concepts:
point
An entity that has a location in space or on a plane, but has no extent
##### Appears in these related concepts:
A quadratic polynomial, function or equation.
##### Appears in these related concepts:
A polynomial equation of the second degree.
##### Appears in these related concepts:
system of equations
A set of equations with multiple variables which can be solved using a specific set of values.
##### Appears in these related concepts:
term
any value (variable or constant) or expression separated from another term by a space or an appropriate character, in an overall expression or table.
##### Appears in these related concepts:
variable
A symbol that represents a quantity in a mathematical expression, as used in many sciences