Glossary

11.5 Solve Systems of Nonlinear Equations

4 min read•june 25, 2024

Nonlinear systems of equations are a crucial topic in algebra. They involve solving multiple equations where at least one isn't linear. These systems pop up in real-world scenarios, from physics to economics, making them super relevant.

Solving nonlinear systems requires different techniques than linear ones. We'll look at graphical methods, substitution, elimination, and more advanced approaches. Understanding these methods will help you tackle complex problems in various fields.

Solving Systems of Nonlinear Equations

Graphical intersection for nonlinear systems

Top images from around the web for Graphical intersection for nonlinear systems

  • Solving a System of Nonlinear Equations Using Substitution | Precalculus II View original

    Is this image relevant?

  • Solve Systems of Nonlinear Equations – Intermediate Algebra View original

    Is this image relevant?

  • Solving a System of Nonlinear Equations Using Substitution | Precalculus II View original

    Is this image relevant?

1 of 3

Top images from around the web for Graphical intersection for nonlinear systems

  • Solving a System of Nonlinear Equations Using Substitution | Precalculus II View original

    Is this image relevant?

  • Solve Systems of Nonlinear Equations – Intermediate Algebra View original

    Is this image relevant?

  • Solving a System of Nonlinear Equations Using Substitution | Precalculus II View original

    Is this image relevant?

1 of 3
  • Identify the equations in the
    • Equations can be in the form of polynomials (quadratic, cubic), exponential functions (y=axy = a^x), or trigonometric functions (sine, cosine, tangent)
  • Graph each equation on the same
    • Use a or software to plot the equations accurately
    • Ensure the graphing window is set appropriately to display all relevant
  • Determine the points of intersection between the graphs
    • These points represent the solutions to the nonlinear system (also known as )
    • Estimate the coordinates of the intersection points if exact values are not easily obtainable
  • Verify the solutions by substituting the coordinates into the original equations
    • Confirm that the coordinates satisfy both equations simultaneously

Substitution methods for nonlinear systems

  • Isolate one variable in terms of the other in one of the equations
    • Choose the equation and variable that will result in the simplest substitution
    • For example, if one equation is y=x2+1y = x^2 + 1, isolate yy in terms of xx
  • Substitute the expression for the isolated variable into the other equation
    • This will result in a single equation with one variable
    • For instance, if the other equation is x+y=5x + y = 5, substituting y=x2+1y = x^2 + 1 yields x+(x2+1)=5x + (x^2 + 1) = 5
  • Solve the resulting equation for the remaining variable
    • Use appropriate algebraic techniques based on the type of equation (, exponential, trigonometric)
    • In the example, solve x+x2+1=5x + x^2 + 1 = 5 for xx using quadratic methods
  • Substitute the value of the solved variable back into the original equation to find the corresponding value of the other variable
    • If x=2x = 2 is a solution, substitute it into y=x2+1y = x^2 + 1 to find y=22+1=5y = 2^2 + 1 = 5
  • Check the solutions by substituting them into both original equations
    • Verify that (2,5)(2, 5) satisfies both y=x2+1y = x^2 + 1 and x+y=5x + y = 5

Elimination techniques for nonlinear systems

  • Multiply one or both equations by appropriate factors to eliminate one variable when the equations are added or subtracted
    • Ensure that the coefficients of the variable to be eliminated are opposites
    • For example, if the equations are x2+y=7x^2 + y = 7 and x2−y=1x^2 - y = 1, multiply the second equation by −1-1 to get −x2+y=−1-x^2 + y = -1
  • Add or subtract the modified equations to eliminate one variable
    • This will result in a single equation with one variable
    • In the example, adding x2+y=7x^2 + y = 7 and −x2+y=−1-x^2 + y = -1 eliminates x2x^2 and yields 2y=62y = 6
  • Solve the resulting equation for the remaining variable
    • Use appropriate techniques based on the type of equation (polynomial, exponential, trigonometric)
    • In the example, solve 2y=62y = 6 for yy to get y=3y = 3
  • Substitute the value of the solved variable back into one of the original equations to find the corresponding value of the other variable
    • Substitute y=3y = 3 into x2+y=7x^2 + y = 7 to get x2+3=7x^2 + 3 = 7, which simplifies to x2=4x^2 = 4, giving x=±2x = \pm 2
  • Verify the solutions by substituting them into both original equations
    • Check that (2,3)(2, 3) and (−2,3)(-2, 3) satisfy both x2+y=7x^2 + y = 7 and x2−y=1x^2 - y = 1

Advanced solution methods for nonlinear systems

  • : Use computational techniques to approximate solutions when analytical methods are not feasible
    • Newton's method: An iterative technique for finding roots of equations
    • Secant method: A variation of Newton's method that doesn't require derivatives
  • : Apply repeated calculations to converge on a solution
    • Fixed-point iteration: Start with an initial guess and refine it through successive approximations
    • Relaxation methods: Adjust solution estimates gradually to satisfy the system equations

Real-world applications of nonlinear systems

  • Identify the variables in the problem and assign them symbols (xx, yy)
    • For example, in a problem involving the height of a projectile, let xx be time and yy be height
  • Write two equations that represent the relationships between the variables in the problem
    • Equations may be based on physical laws (), geometric relationships (), or other constraints
    • In the projectile problem, the equations could be y=−16t2+100ty = -16t^2 + 100t (height) and x=5tx = 5t (horizontal distance)
  • Solve the using graphing, substitution, or elimination methods
    • Choose the most appropriate method based on the complexity of the equations
    • For the projectile problem, substitution is suitable: substitute t=x5t = \frac{x}{5} from the second equation into the first to get y=−16x225+20xy = -\frac{16x^2}{25} + 20x
  • Interpret the solutions in the context of the original problem
    • Determine if the solutions make sense and are feasible given the problem constraints
    • In the projectile problem, the solutions represent the height of the projectile at different horizontal distances
  • Communicate the results clearly, including the problem setup, solution method, and final answer
    • Use proper units (meters for distance, seconds for time) and provide explanations when necessary
    • For example, "The height of the projectile at a horizontal distance of 100 meters is approximately 320 meters, as determined by solving the system of equations using the ."

Key Terms to Review (25)

Algebraic Manipulation: Algebraic manipulation refers to the process of performing various operations and transformations on algebraic expressions to simplify, solve, or rearrange them. It involves the application of rules and properties of algebra to manipulate variables, coefficients, and expressions in order to achieve a desired outcome or solve a problem.
Completing the Square: Completing the square is a technique used to solve quadratic equations by transforming them into a perfect square form. This method involves adding a constant to both sides of the equation to create a perfect square on one side, allowing for easier factorization and solution of the equation.
Conic Section: A conic section is a two-dimensional geometric shape that is formed by the intersection of a plane with a three-dimensional cone. These shapes include circles, ellipses, parabolas, and hyperbolas, and they have important applications in mathematics, science, and engineering.
Coordinate Plane: The coordinate plane, also known as the Cartesian coordinate system, is a two-dimensional graphical representation used to locate and visualize points, lines, and other geometric shapes. It consists of a horizontal x-axis and a vertical y-axis that intersect at a point called the origin, forming a grid-like structure that allows for the precise mapping of coordinates.
Cubic Equation: A cubic equation is a polynomial equation of the third degree, where the highest exponent of the variable is 3. These equations take the general form $ax^3 + bx^2 + cx + d = 0$, where $a$, $b$, $c$, and $d$ are real numbers and $a \neq 0$. Cubic equations are an important class of nonlinear equations that arise in various mathematical and scientific applications.
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 involves manipulating the equations in the system to isolate one variable at a time, ultimately leading to the determination of the values for all the variables in the system.
Exponential Function: An exponential function is a mathematical function in which the independent variable appears as the exponent. These functions exhibit a characteristic growth or decay pattern, where the rate of change is proportional to the current value of the function. Exponential functions are widely used to model a variety of real-world phenomena, from population growth to radioactive decay.
Extraneous Solution: An extraneous solution is a solution to an equation that satisfies the equation algebraically but does not satisfy the original problem statement or the domain restrictions of the equation. These solutions are not considered valid answers to the original problem and must be discarded.
Factoring: Factoring is the process of breaking down a polynomial expression into a product of simpler polynomial expressions. This technique is widely used in various areas of mathematics, including solving equations, simplifying rational expressions, and working with quadratic functions.
Fundamental Theorem of Algebra: The fundamental theorem of algebra states that every non-constant polynomial equation with complex number coefficients has at least one complex number solution. This theorem is a fundamental result in algebra that connects the properties of polynomials to the nature of the complex number system.
Graphing Calculator: A graphing calculator is a type of handheld electronic device that is capable of displaying mathematical functions and their graphs on a screen. It is a powerful tool used in various mathematical and scientific disciplines to visualize and analyze data, solve equations, and explore mathematical concepts.
Iterative Techniques: Iterative techniques are mathematical methods that involve repetitive calculations to find a solution to a problem. These techniques are particularly useful for solving systems of nonlinear equations, where the relationships between variables are complex and cannot be easily solved using traditional algebraic methods.
Nonlinear System: A nonlinear system is a mathematical model in which the variables do not have a linear relationship. Unlike linear systems where the outputs are directly proportional to the inputs, nonlinear systems exhibit complex, unpredictable, and often dynamic behaviors that cannot be easily described by simple linear equations.
Numerical Methods: Numerical methods are mathematical techniques used to solve complex problems that cannot be solved analytically. They involve the use of numerical approximations and algorithms to find approximate solutions to problems, particularly those involving systems of nonlinear equations.
Parabola: A parabola is a curved, U-shaped line or surface that is the graph of a quadratic function. It is one of the fundamental conic sections, along with the circle, ellipse, and hyperbola. Parabolas have many important applications in mathematics, physics, and engineering.
Points of Intersection: Points of intersection refer to the locations where two or more lines, curves, or functions intersect or meet. These points represent the solutions to a system of equations, where the variables take on specific values that satisfy all the equations simultaneously.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers. Polynomials are fundamental in algebra and play a crucial role in various mathematical topics covered in this course.
Projectile Motion: Projectile motion is the motion of an object that is launched or projected into the air and moves solely under the influence of gravity, without any other forces acting upon it. This type of motion is characterized by a parabolic trajectory and is governed by the principles of kinematics and the laws of motion.
Pythagorean Theorem: The Pythagorean Theorem is a fundamental mathematical principle that describes the relationship between the sides of a right triangle. It states that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Quadratic Equation: A quadratic equation is a polynomial equation of the second degree, where the highest exponent of the variable is 2. It takes the general form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. Quadratic equations are fundamental in algebra and have important applications in various fields, including physics, engineering, and economics.
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. These equations represent multiple relationships or constraints that need to be satisfied simultaneously.
Solution Set: The solution set is the set of all values of the variable(s) that satisfy an equation, inequality, or system of equations or inequalities. It represents the collection of all possible solutions to a given mathematical problem.
Substitution Method: The substitution method is a technique used to solve systems of linear equations by isolating one variable in one of the equations and then substituting that expression into the other equation(s) to solve for the remaining variable(s). This method is applicable in various contexts, including solving systems of linear equations with two variables, applications with systems of equations, mixture problems with systems of equations, systems of equations with three variables, solving radical equations, and solving systems of nonlinear equations.
System of Equations: A system of equations is a set of two or more related equations that must be solved simultaneously to find the values of the unknown variables. It is a fundamental concept in mathematics that has applications in various fields, including algebra, physics, and economics.
Trigonometric Function: A trigonometric function is a mathematical function that describes the relationship between the angles and side lengths of a right triangle. These functions, such as sine, cosine, and tangent, are widely used in various fields, including engineering, physics, and mathematics, to analyze and solve problems involving periodic phenomena and angular relationships.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide