11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
2 min read•june 18, 2024
Nonlinear equations and inequalities add complexity to algebraic systems. We'll explore methods like substitution, elimination, and graphing to solve these systems, as well as techniques for interpreting and visualizing their solutions.
Real-world applications of nonlinear systems are diverse, from economics to physics. We'll learn to model problems using nonlinear equations and inequalities, solve them, and interpret the results in practical contexts.
Solving Systems of Nonlinear Equations
Nonlinear equation system solutions
Substitution method
Isolate one variable in an equation (x or y)
Substitute isolated variable expression into the other equation
Solve resulting equation for remaining variable
Substitute solved variable value back into expression to find other variable value
Elimination method
Multiply equation(s) by constant to equalize variable coefficients (x or y)
Add or subtract equations to eliminate one variable
Solve resulting equation for remaining variable
Substitute solved variable value into original equation to find other variable value
Graphical solution
Plot both equations on the same coordinate plane
Identify points of intersection as solutions to the system
Graphing and Interpreting Nonlinear Inequalities
Graphing two-variable nonlinear inequalities
Replace symbol with equal sign and graph resulting equation
Use dashed line for strict inequalities (< or >)
Use solid line for inclusive inequalities (≤ or ≥)
Shade region satisfying inequality
Test point not on line to determine shading side
Shade region containing test point if it satisfies inequality
Shade opposite region if test point doesn't satisfy inequality
Interpretation of nonlinear system solutions
Graphical interpretation
Solution to nonlinear equation system is point(s) where equation graphs intersect
Solution to system is overlapping shaded region
Algebraic interpretation
Verify solution by substituting x and y values into original equations or inequalities
Solution should satisfy all system equations or inequalities (simultaneous equations)
Modeling with Nonlinear Systems
Real-world applications of nonlinear systems
Identify problem variables and constraints
Create nonlinear equation or inequality system representing problem
Solve system using substitution, elimination, or graphing
Interpret solution in original problem context
Check solution feasibility and real-world sense
Application examples
Economics supply and demand curves
Physics projectile trajectories
Biology population growth models
Advanced Nonlinear Systems
Quadratic systems and optimization
Quadratic systems involve equations with squared variables
Optimization problems seek to maximize or minimize a function subject to constraints
Algebraic solution methods may include factoring or using the quadratic formula
Graphical solutions can provide visual representation of optimal points
Key Terms to Review (8)
Circle: A circle is a set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius.
Ellipse: An ellipse is a set of all points in a plane such that the sum of their distances from two fixed points (foci) is constant. It has an oval shape and can be represented by its standard equation.
Feasible region: A feasible region is the set of all possible points that satisfy a system of inequalities. It is typically represented as a shaded area on a graph.
Inequality: Inequality is a mathematical statement that shows the relationship between two expressions that are not equal. It often involves symbols like $\lt$, $\leq$, $\gt$, or $\geq$.
Nonlinear inequality: A nonlinear inequality is an inequality that involves a nonlinear expression, typically involving variables raised to powers other than one or involving products of variables. It defines a range of values that satisfy the inequality.
Parabola: A parabola is a symmetric curve formed by the set of all points in a plane that are equidistant from a given point called the focus and a given line called the directrix. It is commonly represented by the quadratic function $y = ax^2 + bx + c$.
System of nonlinear equations: A system of nonlinear equations consists of two or more equations involving at least one variable raised to a power other than one or involving a product of variables. These systems can be solved using methods such as substitution, elimination, or graphical analysis.
System of nonlinear inequalities: A system of nonlinear inequalities consists of two or more inequalities that contain at least one variable to a power other than one or involve products of variables. Solutions to these systems are regions in the plane where the inequalities overlap.