What Are The Solutions To The Following System?${ \begin{cases} -2x^2 + Y = -5 \ y = -3x^2 + 5 \end{cases} }$A. (0, 2)B. (1, -2)C. { (\sqrt{2}, -1)$}$ And { (- \sqrt{2}, -1)$}$D. { (\sqrt{5}, -10)$}$ And

Mar 8, 2025 by ADMIN 205 views

What are the solutions to the following system?

Introduction

Solving a system of equations is a fundamental concept in mathematics, and it involves finding the values of variables that satisfy all the equations in the system. In this article, we will explore a system of two quadratic equations and find its solutions. The system consists of two equations:

-2x^2 + y = -5 y = -3x^2 + 5

Step 1: Write down the given system of equations

The given system of equations is:

{ \begin{cases} -2x^2 + y = -5 \\ y = -3x^2 + 5 \end{cases} \}

Step 2: Substitute the expression for y from the second equation into the first equation

We can substitute the expression for y from the second equation into the first equation to get:

-2x^2 + (-3x^2 + 5) = -5

Step 3: Simplify the resulting equation

Simplifying the resulting equation, we get:

-5x^2 + 5 = -5

Step 4: Rearrange the equation to get a quadratic equation in x

Rearranging the equation, we get:

-5x^2 = -10

Step 5: Divide both sides by -5 to get x^2 = 2

Dividing both sides by -5, we get:

x^2 = 2

Step 6: Take the square root of both sides to get x = ±√2

Taking the square root of both sides, we get:

x = ±√2

Step 7: Substitute the values of x back into the second equation to get y = -3x^2 + 5

Substituting the values of x back into the second equation, we get:

y = -3(±√2)^2 + 5

Step 8: Simplify the expression for y

Simplifying the expression for y, we get:

y = -3(2) + 5

Step 9: Simplify further to get y = -1

Simplifying further, we get:

y = -1

Step 10: Write down the solutions to the system

The solutions to the system are:

( $\sqrt{2}, -1$ ) and ( $- \sqrt{2}, -1$ )

Conclusion

In this article, we have solved a system of two quadratic equations and found its solutions. The solutions to the system are ( $\sqrt{2}, -1$ ) and ( $- \sqrt{2}, -1$ ). This demonstrates the importance of substitution and simplification in solving systems of equations.

Discussion

The solutions to the system can be verified by substituting them back into the original equations. This is an important step in ensuring the accuracy of the solutions.

Final Answer

The final answer is:

( $\sqrt{2}, -1$ ) and ( $- \sqrt{2}, -1$ )

Comparison with Options

Comparing the solutions with the options, we can see that the correct answer is:

C. ( $\sqrt{2}, -1$ ) and ( $- \sqrt{2}, -1$ )

This option matches the solutions we obtained in the previous steps.

Importance of Solving Systems of Equations

Solving systems of equations is an important concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and economics. It helps us to model real-world problems and find the values of variables that satisfy all the equations in the system.

Real-World Applications

Solving systems of equations has numerous real-world applications, such as:

Modeling population growth and decline
Finding the optimal solution to a problem
Solving optimization problems
Modeling electrical circuits
Modeling mechanical systems

Conclusion

In conclusion, solving systems of equations is an important concept in mathematics, and it has numerous applications in various fields. It helps us to model real-world problems and find the values of variables that satisfy all the equations in the system. The solutions to the system are ( $\sqrt{2}, -1$ ) and ( $- \sqrt{2}, -1$ ), and they can be verified by substituting them back into the original equations.

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it involves finding the values of variables that satisfy all the equations in the system. In this article, we will answer some frequently asked questions about solving systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve the same variables. The variables in the equations are related to each other, and the system of equations is used to find the values of the variables that satisfy all the equations.

Q: How do I solve a system of equations?

A: There are several methods to solve a system of equations, including substitution, elimination, and graphing. The method you choose depends on the type of equations and the number of variables involved.

Q: What is substitution?

A: Substitution is a method of solving a system of equations where you substitute the expression for one variable from one equation into the other equation. This helps to eliminate one of the variables and solve for the other variable.

Q: What is elimination?

A: Elimination is a method of solving a system of equations where you add or subtract the equations to eliminate one of the variables. This helps to solve for the other variable.

Q: What is graphing?

A: Graphing is a method of solving a system of equations where you graph the equations on a coordinate plane and find the point of intersection. This point represents the solution to the system of equations.

Q: How do I know which method to use?

A: The method you choose depends on the type of equations and the number of variables involved. If the equations are linear, you can use the elimination method. If the equations are quadratic, you can use the substitution method. If the equations are complex, you can use the graphing method.

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 solutions to ensure they satisfy both equations
Not using the correct method for the type of equations involved
Not simplifying the equations before solving
Not checking for extraneous solutions

Q: How do I check the solutions to ensure they satisfy both equations?

A: To check the solutions, substitute the values of the variables back into both equations and ensure that they satisfy both equations.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that satisfies one of the equations but not the other. This can occur when the equations are not consistent or when the method used to solve the system is incorrect.

Q: How do I avoid extraneous solutions?

A: To avoid extraneous solutions, ensure that the equations are consistent and that the method used to solve the system is correct. Also, check the solutions to ensure they satisfy both equations.

Q: Can I use technology to solve systems of equations?

A: Yes, you can use technology such as calculators or computer software to solve systems of equations. This can be helpful for complex systems or when the equations are difficult to solve by hand.

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

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

Modeling population growth and decline
Finding the optimal solution to a problem
Solving optimization problems
Modeling electrical circuits
Modeling mechanical systems

Conclusion

In conclusion, solving systems of equations is an important concept in mathematics, and it has numerous applications in various fields. By understanding the different methods of solving systems of equations and avoiding common mistakes, you can solve complex systems and find the values of variables that satisfy all the equations in the system.

Final Answer

The final answer is:

Solving systems of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy all the equations in the system. By understanding the different methods of solving systems of equations and avoiding common mistakes, you can solve complex systems and find the values of variables that satisfy all the equations in the system.