11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
2 min read•june 24, 2024
can be tricky, but they're crucial in math and real life. We'll look at different ways to solve them, like substitution, elimination, and graphing. These methods help us find where equations intersect, giving us solutions.
Nonlinear inequalities are just as important. We'll learn how to graph them and find regions that satisfy multiple inequalities at once. This stuff is super useful for solving real-world problems in business, science, and more.
Solving Systems of Nonlinear Equations
Solving nonlinear equation systems
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- involves isolating one variable in an equation
- Substitute the isolated variable expression into the other equation
- Solve the resulting equation for the remaining variable
- Substitute the solved variable value back into the isolation equation to find the other variable value
- eliminates one variable by manipulating equation coefficients
- Multiply equations by constants to make one variable's coefficients equal in magnitude but opposite in sign
- Add the equations together to eliminate the variable
- Solve the resulting equation for the remaining variable
- Substitute the solved variable value into an original equation to find the other variable value
- involves plotting both equations and identifying
Solutions for nonlinear systems
- Solutions to nonlinear equation systems are where equation graphs intersect
- occurs with parallel or
- One solution occurs when graphs intersect at a single point (x, y)
- Infinitely many solutions occur when graphs are identical
- Solutions to are satisfying all inequalities simultaneously
- Typically shaded on a graph to represent the solution set
Advanced solution methods
- uses iterative approximations to find roots of nonlinear equations
- tracks solution paths as system parameters change
- relates solutions of nonlinear systems to local function behavior
Graphing and Applying Systems of Nonlinear Inequalities
Graphing of nonlinear inequalities
- Graph each inequality separately by replacing the inequality symbol with an equal sign
- Use a for ( or )
- Use a for ( or )
- Shade above the graph for or
- Shade below the graph for or
- Identify the region satisfying all inequalities simultaneously by finding the shaded region intersection
Applications of nonlinear systems
- Identify variables and construct equations or inequalities based on given information
- Solve the nonlinear equation system or graph the nonlinear inequality system
- Interpret the solution in the problem context
- For equations, the solution represents specific values satisfying given conditions (price, quantity)
- For inequalities, the solution represents a range of values satisfying given conditions (minimum/maximum values)
Key Terms to Review (32)
≤ (Less Than or Equal To): The symbol '≤' represents the mathematical concept of 'less than or equal to.' It is used to indicate that a value or quantity is less than or equal to another value or quantity. This term is commonly used in the context of inequalities, where it helps define the range of values that satisfy a given condition.
The understanding of the symbol '≤' is crucial in the study of linear inequalities, absolute value inequalities, and systems of nonlinear equations and inequalities involving two variables.
Bézout's theorem: Bézout's theorem is a fundamental result in abstract algebra that relates the number of solutions to a system of polynomial equations to the degrees of the polynomials involved. It provides a powerful tool for analyzing the behavior of nonlinear systems of equations.
Circle Equation: The circle equation is a mathematical representation of a circle, which is a two-dimensional shape where all points on the circumference are equidistant from the center. The circle equation is used to describe the relationship between the coordinates of points on the circle and the radius of the circle.
Conic Sections: Conic sections are the curves formed by the intersection of a plane with a cone. These curves include the circle, ellipse, parabola, and hyperbola, and they have important applications in various fields, including mathematics, physics, and engineering.
Coordinate Plane Regions: The coordinate plane is a two-dimensional graphical representation of a mathematical space, where the position of a point is defined by its coordinates (x, y). Coordinate plane regions refer to the various areas or sections within this coordinate plane that are defined by the relationships between the x and y coordinates.
Coordinate Points: Coordinate points, also known as ordered pairs, are a way to represent and locate specific positions on a two-dimensional coordinate plane. They consist of an x-coordinate and a y-coordinate that together uniquely identify a point's location.
Dashed Line: A dashed line is a type of graphical representation that consists of a series of short line segments separated by small gaps, often used to denote a boundary or indicate a particular characteristic in various contexts, including the study of systems of nonlinear equations and inequalities with two variables.
Degenerate conic sections: Degenerate conic sections are special cases of conic sections that do not form the usual shapes like ellipses, parabolas, or hyperbolas. They occur when the plane intersects the cone at its vertex or in other ways that produce a single point, a line, or intersecting lines.
Elimination Method: The elimination method, also known as the method of elimination, is a technique used to solve systems of linear equations by systematically eliminating variables to find the unique solution. This method is applicable in the context of various topics, including parametric equations, systems of linear equations in two and three variables, and systems of nonlinear equations and inequalities.
Ellipse inequality: An ellipse inequality describes a region on a coordinate plane where the points satisfy the condition of being inside or outside an ellipse. An ellipse is typically defined by a standard equation in the form of $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} \leq 1$$ for points inside or $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} \geq 1$$ for points outside, where (h, k) is the center, and a and b are the lengths of the semi-major and semi-minor axes. Understanding ellipse inequalities allows for the analysis of geometric relationships and regions formed by quadratic equations in two variables.
Feasible Region: The feasible region is the set of all possible solutions that satisfy the constraints of a system of linear or nonlinear inequalities or equations. It represents the area or space in which all the given conditions or constraints are met, and it is the focus of analysis in optimization problems involving systems of inequalities and equations.
Graphing Method: The graphing method is a technique used to solve systems of nonlinear equations and inequalities with two variables. It involves plotting the equations or inequalities on a coordinate plane and identifying the point(s) of intersection, which represent the solution(s) to the system.
Greater Than or Equal To (≥): The symbol ≥ represents the mathematical concept of 'greater than or equal to.' It is used to compare two values and indicates that the value on the left side of the symbol is greater than or equal to the value on the right side. This term is particularly relevant in the context of linear inequalities and systems of nonlinear equations and inequalities involving two variables.
Homotopy Continuation Method: The homotopy continuation method is a numerical technique used to solve systems of nonlinear equations. It involves transforming a difficult problem into a sequence of simpler problems that can be solved iteratively, ultimately leading to the solution of the original system.
Implicit Function Theorem: The implicit function theorem is a fundamental result in multivariable calculus that describes the behavior of a function implicitly defined by an equation involving multiple variables. It provides a way to analyze and differentiate such functions, which are essential in the study of systems of nonlinear equations and inequalities involving two variables.
Inclusive Inequalities: Inclusive inequalities are mathematical statements that express a range of values where the endpoints are included in the solution set. They use the symbols $\geq$ (greater than or equal to) and $\leq$ (less than or equal to) to represent this inclusion of the boundary values.
Infinite Solutions: Infinite solutions refers to a situation where a system of equations has an unlimited number of solutions that satisfy all the equations in the system. This concept is particularly relevant in the context of systems of linear equations with three variables and systems of nonlinear equations and inequalities with two variables.
Intersection Points: Intersection points refer to the locations where two or more curves, lines, or functions intersect, indicating the points at which they share common coordinates. These points are crucial in the analysis and understanding of systems of nonlinear equations and inequalities involving two variables.
Newton's Method: Newton's Method is an iterative technique used to find the roots or solutions of a nonlinear equation. It is a powerful numerical method that can efficiently approximate the solutions to complex equations that cannot be solved analytically.
No Solution: The term 'no solution' refers to a situation where a system of equations or inequalities has no values for the variables that satisfy all the equations or inequalities simultaneously. In other words, there is no set of values that can be assigned to the variables that make all the expressions in the system true.
Non-Intersecting Graphs: Non-intersecting graphs are a type of graphical representation where two or more lines, curves, or functions do not cross or share any common points on the coordinate plane. This concept is particularly relevant in the context of systems of nonlinear equations and inequalities involving two variables.
Nonlinear Equation Systems: Nonlinear equation systems are mathematical models that consist of two or more equations with two or more variables, where at least one of the equations is nonlinear. These systems do not have a linear relationship between the variables, making them more complex to solve compared to linear equation systems.
Nonlinear Inequality Systems: Nonlinear inequality systems are mathematical models that involve a set of inequalities with variables that have a nonlinear relationship. These systems go beyond the linear relationships found in linear inequality systems, allowing for more complex and realistic representations of real-world problems involving variables with nonlinear dependencies.
Parallel Graphs: Parallel graphs are two or more graphs that have the same shape and orientation, but are shifted horizontally or vertically relative to each other. They represent systems of nonlinear equations or inequalities that have the same general shape but are offset from one another.
Quadratic Equations: A quadratic equation is a polynomial equation of the second degree, where the highest exponent of the variable is 2. These equations take the general form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. Quadratic equations are central to the study of systems of nonlinear equations and the rotation of axes.
Simultaneous Equations: Simultaneous equations are a set of two or more equations that share common variables and must be solved together to find the values of those variables. They are a fundamental concept in algebra and are essential for understanding systems of linear and nonlinear equations.
Solid Line: A solid line is a continuous, unbroken line used to represent various mathematical concepts, particularly in the context of systems of nonlinear equations and inequalities involving two variables.
Solution Region: The solution region, in the context of systems of nonlinear equations and inequalities with two variables, refers to the set of all points (x, y) that satisfy the given system of equations and/or inequalities. It represents the area or region in the coordinate plane where the solutions to the system can be found.
Strict Inequalities: Strict inequalities are mathematical expressions that compare two values using the symbols '<' (less than) or '>' (greater than), indicating that one value is strictly less than or strictly greater than the other. These types of inequalities are commonly encountered in the context of systems of nonlinear equations and inequalities involving two variables.
Substitution method: The substitution method is a technique for solving systems of equations by substituting one equation into another. This transforms the system into a single-variable equation that can be solved more easily.
Substitution Method: The substitution method is a technique used to solve systems of linear equations, systems of nonlinear equations, and other types of equations by substituting one variable in terms of another. This method involves isolating a variable in one equation and then substituting that expression into the other equation(s) to solve for the remaining variable(s).
Unique solution: A unique solution refers to a single, distinct answer to a system of equations where all variables can be solved explicitly, resulting in one point of intersection in a graph. This concept is essential in understanding how various systems behave, especially when analyzing the relationships between multiple variables, whether linear or nonlinear. Identifying a unique solution ensures that the system is consistent and that there is a clear and definitive outcome for the values of the variables involved.