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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.
Solve nonlinear systems of equations graphically and algebraically
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.
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 :
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 x2 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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Source: Boundless. “Nonlinear Systems of Equations and Problem-Solving.” Boundless Algebra. Boundless, 01 Jul. 2015. Retrieved 03 Jul. 2015 from https://www.boundless.com/algebra/textbooks/boundless-algebra-textbook/conic-sections-7/nonlinear-systems-of-equations-and-inequalities-52/nonlinear-systems-of-equations-and-problem-solving-221-11097/