Find The Solutions To The System:${ \begin{array}{l} x 2+y 2=9 \ x+y=3 \end{array} }$A. { (-3,0),(3,0)$}$ B. { (0,3),(3,0)$}$ C. { (9,0),(6,-3)$}$ D. { (0,9),(3,-6)$}$ E. { (-6,3),(3,0)$}$

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Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two equations with two variables, x and y. We will use a step-by-step approach to find the solutions to the system.

The System of Equations

The system of equations we will be solving is:

x2+y2=9x+y=3\begin{array}{l} x^2+y^2=9 \\ x+y=3 \end{array}

Step 1: Understand the Equations

The first equation is a quadratic equation in terms of x and y, while the second equation is a linear equation. We can start by analyzing the second equation, which is a linear equation in terms of x and y.

Step 2: Solve the Linear Equation

We can solve the linear equation by isolating one variable in terms of the other. Let's solve for x in terms of y:

x+y=3⇒x=3−yx+y=3 \Rightarrow x=3-y

Step 3: Substitute into the Quadratic Equation

Now that we have expressed x in terms of y, we can substitute this expression into the quadratic equation:

(3−y)2+y2=9(3-y)^2+y^2=9

Step 4: Expand and Simplify

Expanding and simplifying the equation, we get:

9−6y+y2+y2=9⇒2y2−6y=09-6y+y^2+y^2=9 \Rightarrow 2y^2-6y=0

Step 5: Factor and Solve

Factoring out a 2y from the equation, we get:

2y(y−3)=02y(y-3)=0

This equation has two possible solutions: y=0 or y=3.

Step 6: Find the Corresponding Values of x

Now that we have found the values of y, we can find the corresponding values of x by substituting these values into the expression x=3-y:

  • For y=0, x=3-0=3
  • For y=3, x=3-3=0

Step 7: Check the Solutions

We can check the solutions by substituting them back into the original equations:

  • For x=3 and y=0, we have:
    • x2+y2=32+02=9x^2+y^2=3^2+0^2=9
    • x+y=3+0=3x+y=3+0=3
  • For x=0 and y=3, we have:
    • x2+y2=02+32=9x^2+y^2=0^2+3^2=9
    • x+y=0+3=3x+y=0+3=3

Both solutions satisfy the original equations.

Conclusion

In this article, we have solved a system of two equations with two variables, x and y. We used a step-by-step approach to find the solutions to the system. The solutions to the system are (3,0) and (0,3).

Final Answer

The final answer is:

  • A. (−3,0),(3,0)\boxed{(-3,0),(3,0)} is incorrect.
  • B. (0,3),(3,0)\boxed{(0,3),(3,0)} is correct.
  • C. (9,0),(6,−3)\boxed{(9,0),(6,-3)} is incorrect.
  • D. (0,9),(3,−6)\boxed{(0,9),(3,-6)} is incorrect.
  • E. (−6,3),(3,0)\boxed{(-6,3),(3,0)} is incorrect.

The correct answer is B.

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Introduction

In our previous article, we solved a system of two equations with two variables, x and y. We used a step-by-step approach to find the solutions to the system. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used in solving systems of equations.

Q: What is a system of equations?

A system of equations is a set of two or more equations that involve two or more variables. The goal is to find the values of the variables that satisfy all the equations simultaneously.

A: How do I know if a system of equations has a solution?

A system of equations has a solution if the equations are consistent, meaning that they do not contradict each other. If the equations are inconsistent, there is no solution.

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 eliminate one variable.
  • Elimination method: This method involves adding or subtracting the equations to eliminate one variable.
  • Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
  • Matrix method: This method involves using matrices 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, if the equations are linear, the substitution or elimination method may be the best choice. If the equations are quadratic, the matrix method may be more suitable.

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 for consistency: Make sure that the equations are consistent before solving the system.
  • Not using the correct method: Choose the best method for solving the system based on the type of equations and variables involved.
  • Not checking the solutions: Make sure that the solutions satisfy all the equations in the system.

Q: How do I check if a solution is valid?

To check if a solution is valid, substitute the values of the variables into each equation and check if the equation is satisfied. If the equation is satisfied, the solution is valid.

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

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

  • Physics and engineering: Solving systems of equations is used to model and analyze physical systems, such as motion and forces.
  • Economics: Solving systems of equations is used to model and analyze economic systems, such as supply and demand.
  • Computer science: Solving systems of equations is used in computer graphics and game development.

Conclusion

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

Final Answer

The final answer is:

  • Q: What is a system of equations?
    • A: A system of equations is a set of two or more equations that involve two or more variables.
  • Q: How do I know if a system of equations has a solution?
    • A: A system of equations has a solution if the equations are consistent.
  • Q: What are the different methods for solving systems of equations?
    • A: The different methods for solving systems of equations include substitution, elimination, graphical, and matrix methods.
  • Q: How do I choose the best method for solving a system of equations?
    • A: The best method for solving a system of equations depends on the type of equations and the variables involved.
  • 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 for consistency, not using the correct method, and not checking the solutions.
  • Q: How do I check if a solution is valid?
    • A: To check if a solution is valid, substitute the values of the variables into each equation and check if the equation is satisfied.
  • Q: What are some real-world applications of solving systems of equations?
    • A: Solving systems of equations has many real-world applications, including physics and engineering, economics, and computer science.