What Is One Of The Solutions To The Following System Of Equations?$\[ \begin{array}{l} x^2 + Y^2 = 25 \\ y - 3x = -5 \end{array} \\]A. \[$(-3, 4)\$\] B. \[$(4, -3)\$\] C. \[$(3, 4)\$\] D. \[$(-4, 3)\$\]

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Introduction

Solving a system of equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore one of the solutions to a given system of equations, which consists of a quadratic equation and a linear equation.

The System of Equations

The given system of equations is:

x2+y2=25y−3x=−5\begin{array}{l} x^2 + y^2 = 25 \\ y - 3x = -5 \end{array}

The first equation is a quadratic equation, representing a circle with a radius of 5 units, centered at the origin (0, 0). The second equation is a linear equation, representing a line with a slope of 3 and a y-intercept of -5.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the substitution method.

First, we will solve the second equation for y:

y=3x−5y = 3x - 5

Now, we will substitute this expression for y into the first equation:

x2+(3x−5)2=25x^2 + (3x - 5)^2 = 25

Expanding the squared term, we get:

x2+9x2−30x+25=25x^2 + 9x^2 - 30x + 25 = 25

Combine like terms:

10x2−30x=010x^2 - 30x = 0

Factor out the common term:

10x(x−3)=010x(x - 3) = 0

This gives us two possible solutions:

x=0orx=3x = 0 \quad \text{or} \quad x = 3

Finding the Corresponding y-Values

Now that we have found the possible x-values, we can substitute them into the second equation to find the corresponding y-values.

For x = 0, we get:

y=3(0)−5=−5y = 3(0) - 5 = -5

So, the first solution is (0, -5).

For x = 3, we get:

y=3(3)−5=4y = 3(3) - 5 = 4

So, the second solution is (3, 4).

Checking the Solutions

To verify that these solutions satisfy both equations, we can plug them back into the original equations.

For the first solution (0, -5), we get:

02+(−5)2=25and−5−3(0)=−50^2 + (-5)^2 = 25 \quad \text{and} \quad -5 - 3(0) = -5

Both equations are satisfied.

For the second solution (3, 4), we get:

32+42=25and4−3(3)=−53^2 + 4^2 = 25 \quad \text{and} \quad 4 - 3(3) = -5

Both equations are satisfied.

Conclusion

In this article, we have explored one of the solutions to a given system of equations. We used the substitution method to solve the system and found two possible solutions: (0, -5) and (3, 4). We then verified that these solutions satisfy both equations by plugging them back into the original equations.

Final Answer

The correct answer is:

  • C. (3, 4)

This solution satisfies both equations and is one of the solutions to the given system of equations.

Discussion

This problem is a classic example of a system of equations with a quadratic equation and a linear equation. The substitution method is a powerful tool for solving such systems, and it allows us to find the solutions by substituting one equation into the other.

In this case, we used the substitution method to solve the system and found two possible solutions. We then verified that these solutions satisfy both equations by plugging them back into the original equations.

This problem is a great example of how to use algebraic techniques to solve systems of equations. It requires a good understanding of algebraic manipulations, such as substitution and factoring, as well as the ability to verify solutions by plugging them back into the original equations.

Additional Resources

For more information on solving systems of equations, including quadratic equations and linear equations, please see the following resources:

These resources provide a comprehensive overview of solving systems of equations, including quadratic equations and linear equations, and offer additional practice problems and examples to help reinforce your understanding.

Introduction

Solving systems of equations is a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we explored one of the solutions to a given system of equations, which consisted of a quadratic equation and a linear equation. 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. In other words, it is a collection of equations that are related to each other through the variables.

Q: How do I solve a system of equations?

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

Q: What is the substitution method?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one equation is linear and the other equation is quadratic.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one variable. This method is useful when the equations are linear and have the same coefficient for one variable.

Q: What is the graphing method?

A: The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful when the equations are linear and have a simple graph.

Q: How do I verify the solutions?

A: To verify the solutions, plug the values back into the original equations and check if they satisfy both equations. This ensures that the solutions are correct and not just a coincidence.

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

A: Some common mistakes to avoid include:

  • Not checking the solutions by plugging them back into the original equations
  • Not considering all possible solutions
  • Not using the correct method for the type of equations involved
  • Not simplifying the equations before solving them

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

A: The choice of method depends on the type of equations and the number of variables involved. Consider the following factors:

  • The type of equations: linear, quadratic, or polynomial
  • The number of variables: two or more
  • The complexity of the equations: simple or complex
  • The desired outcome: exact solutions or approximate solutions

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

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

  • Physics and engineering: modeling motion, forces, and energy
  • Economics: modeling supply and demand, cost, and revenue
  • Computer science: modeling algorithms and data structures
  • Biology: modeling population growth and disease spread

Q: How can I practice solving systems of equations?

A: Practice makes perfect! Try the following:

  • Work on sample problems and exercises
  • Use online resources and tutorials
  • Join a study group or find a study partner
  • Take online courses or attend workshops

Conclusion

Solving systems of equations is a fundamental concept in mathematics, and it has numerous real-world applications. By understanding the different methods and techniques, you can solve systems of equations with confidence. Remember to verify the solutions, avoid common mistakes, and choose the correct method for the type of equations involved.

Final Tips

  • Practice regularly to improve your skills
  • Use online resources and tutorials to supplement your learning
  • Join a study group or find a study partner to stay motivated
  • Take online courses or attend workshops to deepen your understanding

By following these tips and practicing regularly, you can become proficient in solving systems of equations and apply your skills to real-world problems.