Solve The System Of Equations.${ \begin{array}{l} y = 5x \ y = X^2 + 4 \end{array} }$A. { (-1,-5)$}$ And { (4,20)$}$ B. { (-2,0)$}$ And { (-2,-10)$}$ C. { (0,4)$}$ And { (1,5)$}$

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Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. In this article, we will focus on solving a system of two equations with two variables. The system of equations we will be solving is:

{ \begin{array}{l} y = 5x \\ y = x^2 + 4 \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 55 and the y-intercept is 00. 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 constants. In this case, the quadratic equation is y=x2+4y = x^2 + 4.

Solving the System of Equations

To solve the system of equations, we need to find the values of xx and yy that satisfy both equations simultaneously. We can do this by substituting the expression for yy from the first equation into the second equation.

{ \begin{array}{l} y = 5x \\ y = x^2 + 4 \end{array} \}

Substituting y=5xy = 5x into the second equation, we get:

{ \begin{array}{l} 5x = x^2 + 4 \end{array} \}

Rearranging the Equation

To solve for xx, we need to rearrange the equation to get all the terms on one side. We can do this by subtracting x2x^2 from both sides and subtracting 44 from both sides.

{ \begin{array}{l} 5x - x^2 = 4 \end{array} \}

Factoring the Equation

We can factor the left-hand side of the equation by grouping the terms.

{ \begin{array}{l} -x(x - 5) = 4 \end{array} \}

Solving for x

To solve for xx, we need to set each factor equal to 00 and solve for xx.

{ \begin{array}{l} -x = 0 \quad \text{or} \quad x - 5 = 0 \end{array} \}

Solving for xx, we get:

{ \begin{array}{l} x = 0 \quad \text{or} \quad x = 5 \end{array} \}

Finding the Corresponding Values of y

Now that we have found the values of xx, we can find the corresponding values of yy by substituting the values of xx into one of the original equations.

For x=0x = 0, we get:

{ \begin{array}{l} y = 5(0) = 0 \end{array} \}

For x=5x = 5, we get:

{ \begin{array}{l} y = 5(5) = 25 \end{array} \}

Checking the Solutions

To check the solutions, we need to substitute the values of xx and yy into both original equations to make sure they are true.

For x=0x = 0 and y=0y = 0, we get:

{ \begin{array}{l} 0 = 5(0) \quad \text{and} \quad 0 = 0^2 + 4 \end{array} \}

For x=5x = 5 and y=25y = 25, we get:

{ \begin{array}{l} 25 = 5(5) \quad \text{and} \quad 25 = 5^2 + 4 \end{array} \}

Conclusion

In this article, we have solved the system of equations using the method of substitution. We have found the values of xx and yy that satisfy both equations simultaneously. We have also checked the solutions to make sure they are true.

Final Answer

The final answer is:

{ \begin{array}{l} x = 0 \quad \text{and} \quad y = 0 \\ x = 5 \quad \text{and} \quad y = 25 \end{array} \}

Discussion

The system of equations we have solved is a quadratic equation and a linear equation. The quadratic equation is y=x2+4y = x^2 + 4 and the linear equation is y=5xy = 5x. We have used the method of substitution to solve the system of equations.

Related Topics

  • Solving quadratic equations
  • Solving linear equations
  • Systems of equations

References

  • [1] "Solving Systems of Equations" by Math Open Reference
  • [2] "Systems of Equations" by Khan Academy

Tags

  • system of equations
  • quadratic equation
  • linear equation
  • substitution method
  • algebra
  • mathematics

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Introduction

In the previous article, we solved the system of equations using the method of substitution. In this article, we will answer some frequently asked questions related to 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 are solved simultaneously to find the values of the variables.

Q: What are the different methods of solving systems of equations?

A: There are several methods of solving systems of equations, including:

  • Substitution method
  • Elimination method
  • Graphical method
  • Matrix method

Q: What is the substitution method?

A: The substitution method is a method of solving systems of equations by substituting the expression for one variable from one equation into the other equation.

Q: What is the elimination method?

A: The elimination method is a method of solving systems of equations by adding or subtracting the equations to eliminate one of the variables.

Q: What is the graphical method?

A: The graphical method is a method of solving systems of equations by graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the matrix method?

A: The matrix method is a method of solving systems of equations by representing the system as a matrix and using row operations to solve for the variables.

Q: How do I choose the method of solving systems of equations?

A: The choice of method depends on the type of equations and the number of variables. For example, the substitution method is often used for systems of linear equations, while the graphical method is often used for systems of quadratic equations.

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 the solutions
  • Not using the correct method for the type of equations
  • Not simplifying the equations before solving
  • Not using the correct order of operations

Q: How do I check the solutions to a system of equations?

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

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 to model the motion of objects
  • Engineering: solving systems of equations to design and optimize systems
  • Economics: solving systems of equations to model economic systems
  • Computer Science: solving systems of equations to solve problems in computer science

Q: How do I practice solving systems of equations?

A: To practice solving systems of equations, try the following:

  • Start with simple systems of equations and gradually move to more complex ones
  • Use online resources and practice problems to help you learn
  • Join a study group or find a study partner to practice with
  • Take online courses or watch video tutorials to learn new methods and techniques

Conclusion

Solving systems of equations is an important skill in mathematics and has many real-world applications. By understanding the different methods of solving systems of equations and practicing regularly, you can become proficient in solving systems of equations.

Final Answer

The final answer is:

  • The substitution method is a method of solving systems of equations by substituting the expression for one variable from one equation into the other equation.
  • The elimination method is a method of solving systems of equations by adding or subtracting the equations to eliminate one of the variables.
  • The graphical method is a method of solving systems of equations by graphing the equations on a coordinate plane and finding the point of intersection.
  • The matrix method is a method of solving systems of equations by representing the system as a matrix and using row operations to solve for the variables.

Discussion

Solving systems of equations is an important skill in mathematics and has many real-world applications. By understanding the different methods of solving systems of equations and practicing regularly, you can become proficient in solving systems of equations.

Related Topics

  • Solving quadratic equations
  • Solving linear equations
  • Systems of equations

References

  • [1] "Solving Systems of Equations" by Math Open Reference
  • [2] "Systems of Equations" by Khan Academy

Tags

  • system of equations
  • quadratic equation
  • linear equation
  • substitution method
  • elimination method
  • graphical method
  • matrix method
  • algebra
  • mathematics