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

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Introduction

Solving a system of equations involves finding the values of variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two equations with two variables. The system consists of two linear equations and one quadratic equation.

The System of Equations

The given system of equations is:

y=x−2{ y = x - 2 }

y=x2−3x+2{ y = x^2 - 3x + 2 }

Step 1: Equating the Two Equations

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

x−2=x2−3x+2{ x - 2 = x^2 - 3x + 2 }

Step 2: Rearranging the Equation

Rearranging the equation to get a quadratic equation in terms of x.

x2−4x=0{ x^2 - 4x = 0 }

Step 3: Factoring the Quadratic Equation

Factoring the quadratic equation to find the values of x.

x(x−4)=0{ x(x - 4) = 0 }

Step 4: Solving for x

Solving for x by setting each factor equal to zero.

x=0{ x = 0 }

x−4=0{ x - 4 = 0 }

x=4{ x = 4 }

Step 5: Finding the Corresponding Values of y

Now that we have the values of x, we can substitute them into one of the original equations to find the corresponding values of y.

For x = 0:

y=0−2{ y = 0 - 2 }

y=−2{ y = -2 }

For x = 4:

y=4−2{ y = 4 - 2 }

y=2{ y = 2 }

Step 6: Checking the Solutions

We need to check if the solutions satisfy both equations.

For (0, -2):

y=0−2{ y = 0 - 2 }

y=−2{ y = -2 }

y=02−3(0)+2{ y = 0^2 - 3(0) + 2 }

y=2{ y = 2 }

This solution does not satisfy both equations.

For (4, 2):

y=4−2{ y = 4 - 2 }

y=2{ y = 2 }

y=42−3(4)+2{ y = 4^2 - 3(4) + 2 }

y=2{ y = 2 }

This solution satisfies both equations.

Conclusion

The solution to the system of equations is (4, 2).

Discussion

The given system of equations is a quadratic equation and a linear equation. We can solve the system by equating the two equations and solving for x. We then substitute the values of x into one of the original equations to find the corresponding values of y. We need to check if the solutions satisfy both equations.

Final Answer

The final answer is (4, 2).

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Introduction

In the previous article, we solved a system of two equations with two variables. The system consisted of two linear equations and one quadratic equation. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on 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 two or more variables. The system can consist of linear equations, quadratic equations, or a combination of both.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use various methods such as substitution, elimination, or graphing. The method you choose 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 relationship.

Q: How do I check if the solutions satisfy both equations?

A: To check if the solutions satisfy both equations, you need to substitute the values of the variables into both equations and verify that the equations hold true.

Q: What if I have a system of three or more equations?

A: If you have a system of three or more equations, you can use methods such as substitution, elimination, or graphing to solve the system. However, the system may have multiple solutions or no solution at all.

Q: Can I use a calculator to solve a system of equations?

A: Yes, you can use a calculator to solve a system of equations. Many calculators have built-in functions to solve systems of equations, such as the "solve" function.

Q: What if I get a complex solution?

A: If you get a complex solution, it means that the system of equations has no real solution. In this case, you can use the imaginary part of the solution to find the values of the variables.

Q: Can I use a system of equations to model real-world problems?

A: Yes, you can use a system of equations to model real-world problems. Systems of equations can be used to represent relationships between variables in fields such as physics, engineering, economics, and more.

Conclusion

Solving systems of equations is an essential skill in mathematics and has many real-world applications. By understanding the different methods and techniques for solving systems of equations, you can tackle a wide range of problems and make informed decisions.

Final Answer

The final answer is (4, 2).