Solve The System Of Equations: Y = X − 3 Y = X 2 − 5 X + 6 \begin{array}{l} y = X - 3 \\ y = X^2 - 5x + 6 \end{array} Y = X − 3 Y = X 2 − 5 X + 6 ​ A. (3, 0)B. (2, 0) And (3, 0)C. (1, -2) And (5, 2)D. (0, -3)

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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, where one equation is linear and the other is quadratic. We will use algebraic methods to find the solutions to the system of equations.

The System of Equations

The system of equations we will be solving is given by:

y=x−3y=x2−5x+6\begin{array}{l} y = x - 3 \\ y = x^2 - 5x + 6 \end{array}

Step 1: Setting the Equations Equal to Each Other

To solve the system of equations, we need to set the two equations equal to each other. This is because both equations are equal to y, so we can set them equal to each other.

x−3=x2−5x+6x - 3 = x^2 - 5x + 6

Step 2: Rearranging the Equation

Next, we need to rearrange the equation to get all the terms on one side of the equation.

x2−5x+6−x+3=0x^2 - 5x + 6 - x + 3 = 0

Step 3: Simplifying the Equation

Now, we can simplify the equation by combining like terms.

x2−6x+9=0x^2 - 6x + 9 = 0

Step 4: Factoring the Quadratic Equation

The equation is a quadratic equation, and we can factor it to find the solutions.

(x−3)2=0(x - 3)^2 = 0

Step 5: Solving for x

Now, we can solve for x by setting the expression inside the parentheses equal to zero.

x−3=0x - 3 = 0

Step 6: Finding the Value of x

Solving for x, we get:

x=3x = 3

Step 7: Finding the Value of y

Now that we have the value of x, we can substitute it into one of the original equations to find the value of y.

y=x−3y = x - 3

y=3−3y = 3 - 3

y=0y = 0

Conclusion

Therefore, the solution to the system of equations is (3, 0).

Discussion

The solution to the system of equations is (3, 0). This means that when x is equal to 3, y is equal to 0. This is the only solution to the system of equations.

Alternative Solutions

However, we can also find alternative solutions by using the quadratic formula to solve the quadratic equation.

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x=−(−6)±(−6)2−4(1)(9)2(1)x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(9)}}{2(1)}

x=6±36−362x = \frac{6 \pm \sqrt{36 - 36}}{2}

x=6±02x = \frac{6 \pm \sqrt{0}}{2}

x=62x = \frac{6}{2}

x=3x = 3

Conclusion

Therefore, the alternative solution to the system of equations is also (3, 0).

Final Answer

The final answer to the system of equations is (3, 0).

Discussion Category

This problem falls under the category of mathematics, specifically algebra.

Related Problems

This problem is related to other problems in algebra, such as solving quadratic equations and systems of linear equations.

Key Concepts

The key concepts in this problem are:

  • Solving quadratic equations
  • Systems of linear equations
  • Algebraic methods

Conclusion

In conclusion, solving a system of equations involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we solved a system of two equations, where one equation is linear and the other is quadratic. We used algebraic methods to find the solutions to the system of equations. The final answer to the system of equations is (3, 0).

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Introduction

In our previous article, we solved a system of two equations, where one equation is linear and the other is quadratic. We used algebraic methods to find the solutions to the system of equations. 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 are related to each other. In this article, we solved a system of two equations, where one equation is linear and the other is quadratic.

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

A: A linear equation is an equation in which the highest power of the variable is 1. For example, y = x - 3 is a linear equation. A quadratic equation is an equation in which the highest power of the variable is 2. For example, y = x^2 - 5x + 6 is a quadratic equation.

Q: How do I solve a system of equations?

A: To solve a system of equations, you need to set the two equations equal to each other and then solve for the variable. In this article, we used algebraic methods to solve the system of equations.

Q: What are the steps to solve a system of equations?

A: The steps to solve a system of equations are:

  1. Set the two equations equal to each other.
  2. Rearrange the equation to get all the terms on one side of the equation.
  3. Simplify the equation by combining like terms.
  4. Factor the quadratic equation, if possible.
  5. Solve for the variable.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. In this article, we used the quadratic formula to solve the quadratic equation.

Q: What are the advantages and disadvantages of using the quadratic formula?

A: The advantages of using the quadratic formula are:

  • It is a general method that can be used to solve any quadratic equation.
  • It is a quick and easy method to solve quadratic equations.

The disadvantages of using the quadratic formula are:

  • It can be difficult to use if the equation is complex.
  • It can be time-consuming to plug in the values of a, b, and c into the formula.

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

A: Some common mistakes to avoid when solving a system of equations are:

  • Not setting the two equations equal to each other.
  • Not rearranging the equation to get all the terms on one side of the equation.
  • Not simplifying the equation by combining like terms.
  • Not factoring the quadratic equation, if possible.
  • Not solving for the variable.

Q: How do I check my answer?

A: To check your answer, you need to plug the values of x and y back into the original equations to make sure they are true.

Conclusion

In conclusion, solving a system of equations involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we provided a Q&A guide to help you understand the concepts and methods used to solve the system of equations. We also discussed the advantages and disadvantages of using the quadratic formula and some common mistakes to avoid when solving a system of equations.

Final Answer

The final answer to the system of equations is (3, 0).

Discussion Category

This problem falls under the category of mathematics, specifically algebra.

Related Problems

This problem is related to other problems in algebra, such as solving quadratic equations and systems of linear equations.

Key Concepts

The key concepts in this problem are:

  • Solving quadratic equations
  • Systems of linear equations
  • Algebraic methods
  • Quadratic formula

Conclusion

In conclusion, solving a system of equations is an important concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. We hope that this Q&A guide has helped you understand the concepts and methods used to solve the system of equations.