Find All Solutions Of The System:$\[ \left\{ \begin{array}{l} y = 100 - X^2 \\ y = X^2 - 100 \end{array} \right. \\]The Two Solutions Of The System Are:- The One With \[$x \ \textless \ 0\$\] Is: $\[ \begin{array}{l} X =

by ADMIN 228 views

===========================================================

Introduction

In mathematics, solving a system of equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a specific system of equations involving quadratic functions. We will break down the problem step by step, providing a clear and concise explanation of the solution process.

The System of Equations

The system of equations we will be solving is given by:

{y=100−x2y=x2−100\left\{ \begin{array}{l} y = 100 - x^2 \\ y = x^2 - 100 \end{array} \right.

This system consists of two equations, both of which are quadratic functions of the variable xx. Our goal is to find the values of xx that satisfy both equations simultaneously.

Setting Up the Equations

To solve the system of equations, we need to set up the equations in a way that allows us to easily compare and contrast them. We can do this by rewriting the second equation in terms of yy:

y=x2−100y = x^2 - 100

Now, we can see that both equations are equal to yy, so we can set them equal to each other:

100−x2=x2−100100 - x^2 = x^2 - 100

Simplifying the Equation

To simplify the equation, we can combine like terms:

200=2x2200 = 2x^2

Solving for xx

Now, we can solve for xx by dividing both sides of the equation by 2:

x2=100x^2 = 100

Taking the square root of both sides, we get:

x=±10x = \pm 10

The Two Solutions

We have found two possible values for xx: x=10x = 10 and x=−10x = -10. However, we need to check which of these solutions satisfy the original system of equations.

Checking the Solutions

To check the solutions, we can plug each value of xx back into the original equations and see if they satisfy both equations simultaneously.

For x=10x = 10, we get:

y=100−102=100−100=0y = 100 - 10^2 = 100 - 100 = 0

y=102−100=100−100=0y = 10^2 - 100 = 100 - 100 = 0

Since both equations are satisfied, x=10x = 10 is a valid solution.

For x=−10x = -10, we get:

y=100−(−10)2=100−100=0y = 100 - (-10)^2 = 100 - 100 = 0

y=(−10)2−100=100−100=0y = (-10)^2 - 100 = 100 - 100 = 0

Since both equations are satisfied, x=−10x = -10 is also a valid solution.

Conclusion

In this article, we have solved a system of equations involving quadratic functions. We have found two possible values for xx: x=10x = 10 and x=−10x = -10. We have checked these solutions by plugging them back into the original equations and found that both solutions satisfy both equations simultaneously.

Discussion

The system of equations we solved in this article is a classic example of a quadratic system. Quadratic systems are a fundamental concept in mathematics, and they have many real-world applications. For example, quadratic systems can be used to model the motion of objects under the influence of gravity, or to optimize the design of a physical system.

Final Thoughts

In conclusion, solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. By following a step-by-step approach, we can solve even the most complex systems of equations. Whether you are a student or a professional, understanding how to solve systems of equations is an essential skill that can be applied to a wide range of real-world problems.

Additional Resources

Related Topics

Glossary

  • System of Equations: A set of equations that involve multiple variables and are used to solve for the values of those variables.
  • Quadratic Function: A polynomial function of degree two, which can be written in the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c.
  • Linear Algebra: A branch of mathematics that deals with the study of linear equations, vector spaces, and linear transformations.
  • Calculus: A branch of mathematics that deals with the study of rates of change and accumulation.

=====================================================

Introduction

In our previous article, we solved a system of equations involving quadratic functions. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of equations that involve multiple variables and are used to solve for the values of those variables.

Q: What are the different types of systems of equations?

A: There are two main types of systems of equations: linear systems and nonlinear systems. Linear systems involve linear equations, while nonlinear systems involve nonlinear equations.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use the following steps:

  1. Write down the equations: Write down the equations in the system.
  2. Solve for one variable: Solve for one variable in terms of the other variables.
  3. Substitute the expression: Substitute the expression for the variable into one of the original equations.
  4. Solve for the other variable: Solve for the other variable.
  5. Check the solution: Check the solution by plugging it back into the original equations.

Q: What is the difference between a linear system and a nonlinear system?

A: A linear system involves linear equations, which can be written in the form ax+by=cax + by = c. A nonlinear system involves nonlinear equations, which cannot be written in this form.

Q: How do I determine if a system of equations is linear or nonlinear?

A: To determine if a system of equations is linear or nonlinear, you can check if the equations can be written in the form ax+by=cax + by = c. If they can, then the system is linear. If they cannot, then the system is nonlinear.

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

A: Some common techniques for solving systems of equations include:

  • Substitution method: Substitute the expression for one variable into one of the original equations.
  • Elimination method: Add or subtract the equations to eliminate one of the variables.
  • Graphical method: Plot the equations on a graph and find the point of intersection.

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

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

  • Physics: Systems of equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Systems of equations are used to design and optimize physical systems.
  • Economics: Systems of equations are used to model economic systems and make predictions about economic trends.

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

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

  • Not checking the solution: Make sure to check the solution by plugging it back into the original equations.
  • Not using the correct method: Make sure to use the correct method for the type of system you are solving.
  • Not being careful with algebraic manipulations: Make sure to be careful with algebraic manipulations and avoid making mistakes.

Conclusion

In this article, we have provided a Q&A guide to help you better understand the concepts and techniques involved in solving systems of equations. We have covered topics such as the definition of a system of equations, the different types of systems of equations, and common techniques for solving systems of equations. We have also discussed real-world applications of systems of equations and common mistakes to avoid when solving systems of equations.

Additional Resources

Related Topics

Glossary

  • System of Equations: A set of equations that involve multiple variables and are used to solve for the values of those variables.
  • Quadratic Function: A polynomial function of degree two, which can be written in the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c.
  • Linear Algebra: A branch of mathematics that deals with the study of linear equations, vector spaces, and linear transformations.
  • Calculus: A branch of mathematics that deals with the study of rates of change and accumulation.