The System of Nonlinear Equations

Oct 13, 2025 by windailys.com 53 views

Introduction

In mathematics, a system of nonlinear equations is a set of equations that involve nonlinear functions. Solving such systems can be challenging, and it often requires advanced mathematical techniques. In this article, we will explore a specific system of nonlinear equations and find its solution.

The System of Nonlinear Equations

The given system of nonlinear equations is:

\begin{cases} y = x^2 - 3 \ y = \sqrt{2x} - 1 \end{cases}

Our goal is to find the values of x and y that satisfy both equations simultaneously.

Equating the Two Equations

To solve this system, we can equate the two equations and solve for x. This is because both equations are equal to y, so we can set them equal to each other:

x^2 - 3 = \sqrt{2x} - 1

Simplifying the Equation

To simplify the equation, we can start by isolating the square root term:

x^2 - 3 + 1 = \sqrt{2x}

This simplifies to:

x^2 - 2 = \sqrt{2x}

Squaring Both Sides

To eliminate the square root, we can square both sides of the equation:

(x^2 - 2)^2 = 2x

Expanding the Left Side

Expanding the left side of the equation, we get:

x^4 - 4x^2 + 4 = 2x

Rearranging the Terms

Rearranging the terms, we get:

x^4 - 4x^2 - 2x + 4 = 0

Factoring the Quadratic

This is a quartic equation, but we can try to factor it as a quadratic in x^2:

(x^2 - 2)^2 - 2(x^2 - 2) + 4 = 0

Simplifying the Factored Form

Simplifying the factored form, we get:

(x^2 - 2)^2 - 2(x^2 - 2) + 4 = (x^2 - 2)(x^2 - 2 - 2) + 4 = (x^2 - 2)(x^2 - 4) + 4

Solving for x

Now we can solve for x by setting each factor equal to zero:

x^2 - 2 = 0 or x^2 - 4 = 0

Solving the First Equation

Solving the first equation, we get:

x^2 = 2

x = ±√2

Solving the Second Equation

Solving the second equation, we get:

x^2 = 4

x = ±2

Checking the Solutions

We need to check each solution to see if it satisfies both original equations.

Checking x = √2

Substituting x = √2 into the first equation, we get:

y = (√2)^2 - 3

y = 2 - 3

y = -1

Substituting x = √2 into the second equation, we get:

y = √(2(√2)) - 1

y = √(2√2) - 1

y = √(2√2) - 1 ≠ -1

So x = √2 is not a solution.

Checking x = -√2

Substituting x = -√2 into the first equation, we get:

y = (-√2)^2 - 3

y = 2 - 3

y = -1

Substituting x = -√2 into the second equation, we get:

y = √(2(-√2)) - 1

y = √(-2√2) - 1

y = √(-2√2) - 1 ≠ -1

So x = -√2 is not a solution.

Checking x = 2

Substituting x = 2 into the first equation, we get:

y = (2)^2 - 3

y = 4 - 3

y = 1

Substituting x = 2 into the second equation, we get:

y = √(2(2)) - 1

y = √(4) - 1

y = 2 - 1

y = 1

So x = 2 is a solution.

Checking x = -2

Substituting x = -2 into the first equation, we get:

y = (-2)^2 - 3

y = 4 - 3

y = 1

Substituting x = -2 into the second equation, we get:

y = √(2(-2)) - 1

y = √(-4) - 1

y = √(-4) - 1 ≠ 1

So x = -2 is not a solution.

Conclusion

In conclusion, the solution to the system of nonlinear equations is x = 2 and y = 1.

Answer

The correct answer is:

C. ${$ (2, 1)$}$

Discussion

Q: What is a system of nonlinear equations?

A: A system of nonlinear equations is a set of equations that involve nonlinear functions. Nonlinear functions are functions that do not have a constant rate of change, and they can be represented by equations that are not linear.

Q: How do you solve a system of nonlinear equations?

A: Solving a system of nonlinear equations can be challenging, and it often requires advanced mathematical techniques. One common method is to equate the two equations and simplify the resulting equation. From there, we can use algebraic techniques to solve for x and then check the solutions to see if they satisfy both original equations.

Q: What are some common methods for solving systems of nonlinear equations?

A: Some common methods for solving systems of nonlinear equations include:

Equating the two equations and simplifying the resulting equation
Using algebraic techniques to solve for x
Checking the solutions to see if they satisfy both original equations
Using numerical methods, such as the Newton-Raphson method, to approximate the solutions

Q: What are some common types of nonlinear equations?

A: Some common types of nonlinear equations include:

Quadratic equations: equations that involve a squared variable, such as x^2
Polynomial equations: equations that involve a variable raised to a power, such as x^3
Rational equations: equations that involve a fraction, such as x/2
Trigonometric equations: equations that involve trigonometric functions, such as sin(x)

Q: How do you check the solutions to a system of nonlinear equations?

A: To check the solutions to a system of nonlinear equations, we need to substitute the solutions into both original equations and see if they are satisfied. If the solutions satisfy both equations, then they are valid solutions.

Q: What are some common mistakes to avoid when solving systems of nonlinear equations?

A: Some common mistakes to avoid when solving systems of nonlinear equations include:

Not equating the two equations and simplifying the resulting equation
Not checking the solutions to see if they satisfy both original equations
Not using algebraic techniques to solve for x
Not using numerical methods, such as the Newton-Raphson method, to approximate the solutions

Q: How do you graph a system of nonlinear equations?

A: To graph a system of nonlinear equations, we can use a graphing calculator or a computer program to plot the two equations on the same coordinate plane. We can then identify the points of intersection, which represent the solutions to the system.

Q: What are some real-world applications of systems of nonlinear equations?

A: Systems of nonlinear equations have many real-world applications, including:

Physics: to model the motion of objects under the influence of gravity or other forces
Engineering: to design and optimize systems, such as bridges or buildings
Economics: to model the behavior of economic systems, such as supply and demand
Biology: to model the growth and behavior of populations

Q: How do you use systems of nonlinear equations in real-world applications?

A: To use systems of nonlinear equations in real-world applications, we need to:

Identify the variables and parameters involved in the problem
Write down the equations that describe the problem
Solve the system of equations using algebraic or numerical methods
Interpret the results in the context of the problem

Conclusion

In conclusion, systems of nonlinear equations are a fundamental concept in mathematics and have many real-world applications. By understanding how to solve and graph these systems, we can apply them to a wide range of problems in physics, engineering, economics, and biology.