Find The Solution To The System Of Equations. Round To The Nearest Tenth If Necessary.$\begin{array}{l} F(x)=x^2-5 \\ G(x)=x-1 \end{array}$

by ADMIN 140 views

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

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. In this article, we will focus on solving a system of equations using the given functions f(x) and g(x). We will use algebraic methods to find the solution and round the answer to the nearest tenth if necessary.

Understanding the Functions

The given functions are f(x) = x^2 - 5 and g(x) = x - 1. These are quadratic and linear functions, respectively. To solve the system of equations, we need to find the values of x that satisfy both equations.

Graphical Representation

To visualize the problem, let's graph the functions f(x) and g(x) on the same coordinate plane.

import numpy as np
import matplotlib.pyplot as plt

# Define the functions
def f(x):
    return x**2 - 5

def g(x):
    return x - 1

# Generate x values
x = np.linspace(-10, 10, 400)

# Calculate y values
y1 = f(x)
y2 = g(x)

# Create the plot
plt.plot(x, y1, label='f(x) = x^2 - 5')
plt.plot(x, y2, label='g(x) = x - 1')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of f(x) and g(x)')
plt.legend()
plt.grid(True)
plt.show()

Solving the System of Equations

To solve the system of equations, we need to find the values of x that satisfy both equations. We can do this by setting the two equations equal to each other and solving for x.

f(x) = g(x)

x^2 - 5 = x - 1

x^2 - x - 4 = 0

Now, we need to solve the quadratic equation x^2 - x - 4 = 0.

Factoring the Quadratic Equation

We can factor the quadratic equation x^2 - x - 4 = 0 as follows:

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

This gives us two possible solutions: x - 2 = 0 and x + 2 = 0.

Solving for x

Solving for x, we get:

x - 2 = 0 --> x = 2

x + 2 = 0 --> x = -2

Conclusion

In this article, we solved a system of equations using the given functions f(x) and g(x). We used algebraic methods to find the values of x that satisfy both equations and rounded the answer to the nearest tenth if necessary. The solution to the system of equations is x = 2 and x = -2.

Final Answer

The final answer is 2\boxed{2} and −2\boxed{-2}.

Additional Resources

For more information on solving systems of equations, please refer to the following resources:

Note: The final answer is in a boxed format to indicate that it is the solution to the problem. The additional resources section provides links to external websites that offer more information on the topic.

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

Introduction

In our previous article, we discussed how to solve a system of equations using algebraic methods. In this article, we will answer some frequently asked questions about solving systems of equations.

Q: What is a system of equations?

A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: How do I know if I have a system of equations?

You have a system of equations if you have two or more equations with two or more variables. For example:

f(x) = x^2 - 5 g(x) = x - 1

These are two equations with one variable, x.

Q: What are the different methods for solving systems of equations?

There are several methods for solving systems of equations, including:

  • Substitution Method: This method involves substituting one equation into the other equation to solve for the variable.
  • Elimination Method: This method involves adding or subtracting the equations to eliminate one of the variables.
  • Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
  • Algebraic Method: This method involves using algebraic techniques, such as factoring and solving quadratic equations, to solve the system of equations.

Q: How do I choose the best method for solving a system of equations?

The best method for solving a system of equations depends on the type of equations and the variables involved. For example:

  • Substitution Method: This method is best used when one equation is linear and the other equation is quadratic.
  • Elimination Method: This method is best used when the equations are linear and have the same variable.
  • Graphical Method: This method is best used when the equations are linear and have two variables.
  • Algebraic Method: This method is best used when the equations are quadratic and have two variables.

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

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: Choose the best method for solving the system of equations.
  • Not simplifying the equations: Simplify the equations before solving them.
  • Not checking for extraneous solutions: Check for extraneous solutions by plugging the solution back into the original equations.

Q: How do I check my solution to a system of equations?

To check your solution to a system of equations, plug the solution back into the original equations and make sure it satisfies both equations.

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

Solving systems of equations has many real-world applications, including:

  • Physics: Solving systems of equations is used to model the motion of objects and predict their behavior.
  • Engineering: Solving systems of equations is used to design and optimize systems, such as bridges and buildings.
  • Economics: Solving systems of equations is used to model economic systems and predict the behavior of markets.
  • Computer Science: Solving systems of equations is used to develop algorithms and solve problems in computer science.

Conclusion

In this article, we answered some frequently asked questions about solving systems of equations. We discussed the different methods for solving systems of equations and provided some tips for choosing the best method. We also discussed some common mistakes to avoid when solving systems of equations and how to check your solution.

Final Answer

The final answer is that solving systems of equations is a powerful tool for solving problems in mathematics and real-world applications.

Additional Resources

For more information on solving systems of equations, please refer to the following resources: