Solve The System Of Equations:${ \begin{array}{l} y = X - 2 \ y = X^2 - 3x + 2 \end{array} }$A. (2, 0)B. (1, -1) And (3, 1)C. (0, -2)D. (2, 0) And (1, 0)

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Introduction

Solving a system of equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on solving a system of two equations with two variables. We will use algebraic methods to find the solutions to the system.

The System of Equations

The system of equations we will be solving is:

y=xβˆ’2y=x2βˆ’3x+2\begin{array}{l} y = x - 2 \\ y = x^2 - 3x + 2 \end{array}

Step 1: Equating the Two Equations

To solve the system of equations, we can start by equating the two equations. This means that we set the two equations equal to each other:

xβˆ’2=x2βˆ’3x+2x - 2 = x^2 - 3x + 2

Step 2: Rearranging the Equation

Next, we can rearrange the equation to get a quadratic equation in terms of x:

x2βˆ’4x+4=0x^2 - 4x + 4 = 0

Step 3: Factoring the Quadratic Equation

We can factor the quadratic equation as:

(xβˆ’2)2=0(x - 2)^2 = 0

Step 4: Solving for x

Now, we can solve for x by setting the factored expression equal to zero:

xβˆ’2=0x - 2 = 0

x=2x = 2

Step 5: Finding the Corresponding y-Value

Now that we have found the value of x, we can substitute it into one of the original equations to find the corresponding y-value. Let's use the first equation:

y=xβˆ’2y = x - 2

y=2βˆ’2y = 2 - 2

y=0y = 0

Step 6: Checking the Solution

We have found one solution to the system of equations: (2, 0). However, we need to check if this solution satisfies both equations. Let's plug in the values of x and y into both equations:

y=xβˆ’2y = x - 2

0=2βˆ’20 = 2 - 2

0=00 = 0

This is true, so the solution (2, 0) satisfies both equations.

Conclusion

In this article, we have solved a system of two equations with two variables using algebraic methods. We have found one solution to the system: (2, 0). We have also checked that this solution satisfies both equations.

Final Answer

The final answer is (2, 0).

Discussion

The system of equations we solved is a quadratic equation in terms of x. We used algebraic methods to find the solutions to the system. The solution we found is (2, 0). We also checked that this solution satisfies both equations.

Related Topics

  • Solving systems of linear equations
  • Solving quadratic equations
  • Algebraic methods for solving systems of equations

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by James Stewart
  • [3] "Linear Algebra and Its Applications" by Gilbert Strang

Additional Resources

  • Khan Academy: Solving Systems of Equations
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram Alpha: Solving Systems of Equations

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Introduction

In our previous article, we solved a system of two equations with two variables using algebraic methods. We found one solution to the system: (2, 0). In this article, we will provide a Q&A guide to help you understand the concepts and methods used to solve the system of equations.

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. In our case, we had two equations with two variables: x and y.

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

A: To determine if a system of equations has a solution, we need to check if the two equations are consistent. If the two equations are consistent, then the system has a solution. If the two equations are inconsistent, then the system does not have a solution.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, the equation y = x - 2 is a linear equation. A quadratic equation is an equation in which the highest power of the variable is 2. For example, the equation y = x^2 - 3x + 2 is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, we can use factoring, the quadratic formula, or other algebraic methods. In our case, we used factoring to solve the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. The formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: How do I check if a solution satisfies both equations?

A: To check if a solution satisfies both equations, we need to plug in the values of x and y into both equations and check if the equations are true.

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 if the two equations are consistent
  • Not using the correct algebraic methods to solve the system
  • Not checking if the solution satisfies both equations

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

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

  • Physics: Solving systems of equations is used to model the motion of objects and to solve problems involving forces and energies.
  • Engineering: Solving systems of equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Solving systems of equations is used to model economic systems and to solve problems involving supply and demand.

Conclusion

In this article, we have provided a Q&A guide to help you understand the concepts and methods used to solve the system of equations. We have also discussed some common mistakes to avoid and some real-world applications of solving systems of equations.

Final Answer

The final answer is (2, 0).

Discussion

The system of equations we solved is a quadratic equation in terms of x. We used algebraic methods to find the solutions to the system. The solution we found is (2, 0). We also checked that this solution satisfies both equations.

Related Topics

  • Solving systems of linear equations
  • Solving quadratic equations
  • Algebraic methods for solving systems of equations

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by James Stewart
  • [3] "Linear Algebra and Its Applications" by Gilbert Strang

Additional Resources

  • Khan Academy: Solving Systems of Equations
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram Alpha: Solving Systems of Equations