Solve The System Of Equations: Y = 2 X − 3 Y = X 2 − 3 \begin{array}{l} Y = 2x - 3 \\ Y = X^2 - 3 \end{array} Y = 2 X − 3 Y = X 2 − 3 ​ A. { (0, 3)$}$ And { (2, 0)$}$B. { (0, -3)$}$ And { (2, 1)$}$C. { (-1, -5)$}$ And { (4, 5)$}$D.

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Introduction

In mathematics, solving a system of equations 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. The given system of equations is:

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

Understanding the System of Equations

The first equation is a linear equation in the form of y=mx+by = mx + b, where mm is the slope and bb is the y-intercept. In this case, the slope is 22 and the y-intercept is −3-3. The second equation is a quadratic equation in the form of y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are coefficients. In this case, the coefficient of x2x^2 is 11, the coefficient of xx is 00, and the constant term is −3-3.

Solving the System of Equations

To solve the system of equations, we can use the substitution method. We can substitute the expression for yy from the first equation into the second equation. This will give us an equation with only one variable, which we can solve.

Substituting y=2x−3y = 2x - 3 into the second equation, we get:

2x−3=x2−32x - 3 = x^2 - 3

Rearranging the Equation

To make it easier to solve, we can rearrange the equation by moving all terms to one side:

x2−2x=0x^2 - 2x = 0

Factoring the Equation

We can factor out an xx from the left-hand side of the equation:

x(x−2)=0x(x - 2) = 0

Solving for x

To find the values of xx, we can set each factor equal to zero and solve for xx:

x=0x = 0 or x−2=0x - 2 = 0

Solving for xx, we get:

x=0x = 0 or x=2x = 2

Finding the Corresponding Values of y

Now that we have found the values of xx, we can substitute them into one of the original equations to find the corresponding values of yy. We will use the first equation:

y=2x−3y = 2x - 3

Substituting x=0x = 0, we get:

y=2(0)−3=−3y = 2(0) - 3 = -3

Substituting x=2x = 2, we get:

y=2(2)−3=1y = 2(2) - 3 = 1

Conclusion

Therefore, the solutions to the system of equations are (0,−3)(0, -3) and (2,1)(2, 1).

Answer

The correct answer is:

B. {(0, -3)$}$ and {(2, 1)$}$

Discussion

Solving a system of equations involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we used the substitution method to solve a system of two equations with two variables. We rearranged the equation, factored it, and solved for xx. Then, we found the corresponding values of yy by substituting the values of xx into one of the original equations. The solutions to the system of equations are (0,−3)(0, -3) and (2,1)(2, 1).

Final Thoughts

Solving a system of equations is an important concept in mathematics, and it has many real-world applications. In this article, we used a step-by-step approach to solve a system of two equations with two variables. We hope that this article has provided a clear and concise explanation of how to solve a system of equations.

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Introduction

In our previous article, we solved a system of two equations with two variables using the substitution method. In this article, we will provide a Q&A guide to help you understand the concept of solving a 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. The goal is to find the values of the variables that satisfy all the equations simultaneously.

Q: What are the different methods for solving a system of equations?

A: There are several methods for solving a system of equations, including:

  • Substitution method: This involves substituting the expression for one variable from one equation into the other equation.
  • Elimination method: This involves adding or subtracting the equations to eliminate one of the variables.
  • Graphical method: This involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method involves substituting the expression for one variable from one equation into the other equation. This is the method we used in our previous article.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of one variable are the same in both equations.

Q: What is the graphical method?

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

Q: How do I know which method to use?

A: The choice of method depends on the type of equations and the variables involved. If the equations are linear, the graphical method may be the easiest to use. If the equations are quadratic, the substitution method may be more suitable.

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 include:

  • Not checking the solutions to make sure they satisfy both equations.
  • Not using the correct method for the type of equations involved.
  • Not simplifying the equations before solving.

Q: How do I check my solutions?

A: To check your solutions, substitute the values of the variables into both equations and make sure they are true.

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

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

  • Physics: Solving a system of equations can help model the motion of objects.
  • Engineering: Solving a system of equations can help design and optimize systems.
  • Economics: Solving a system of equations can help model economic systems.

Conclusion

Solving a system of equations is an important concept in mathematics, and it has many real-world applications. In this article, we provided a Q&A guide to help you understand the concept of solving a system of equations. We hope that this article has provided a clear and concise explanation of how to solve a system of equations.

Final Thoughts

Solving a system of equations is a fundamental concept in mathematics, and it has many real-world applications. We hope that this article has provided a helpful guide for you to understand the concept of solving a system of equations.

Additional Resources

For more information on solving a system of equations, we recommend the following resources:

  • Khan Academy: Solving Systems of Equations
  • Mathway: Solving Systems of Equations
  • Wolfram Alpha: Solving Systems of Equations

FAQs

Q: What is the difference between a system of equations and a system of inequalities?

A: A system of equations involves two or more equations that involve two or more variables, while a system of inequalities involves two or more inequalities that involve two or more variables.

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. However, it's always a good idea to check your solutions by hand to make sure they are correct.

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

A: Yes, you can use a computer program to solve a system of equations. Many computer programs, such as Mathematica and MATLAB, have built-in functions for solving systems of equations.