What Is The Solution To This System Of Equations?${ \left{ \begin{array}{c} 2x + 2y = 12 \ x + Y = 6 \end{array} \right. }$

by ADMIN 125 views

Introduction

System of Equations: 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 linear equations with two variables. The system of equations is given as:

{ \left\{ \begin{array}{c} 2x + 2y = 12 \\ x + y = 6 \end{array} \right. \}

Understanding the System of Equations

The system of equations consists of two linear equations with two variables, x and y. The first equation is 2x + 2y = 12, and the second equation is x + y = 6. To solve this system of equations, we need to find the values of x and y that satisfy both equations simultaneously.

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Let's solve the second equation for y:

y = 6 - x

Now, substitute this expression for y into the first equation:

2x + 2(6 - x) = 12

Expand and simplify the equation:

2x + 12 - 2x = 12

Combine like terms:

12 = 12

This equation is true for all values of x. Therefore, we can conclude that x can be any real number.

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. This method involves adding or subtracting the equations to eliminate one variable.

Let's multiply the second equation by 2 to make the coefficients of y in both equations equal:

2(x + y) = 2(6)

Expand and simplify the equation:

2x + 2y = 12

Now, subtract the first equation from this new equation:

(2x + 2y) - (2x + 2y) = 12 - 12

This equation is true for all values of x and y. Therefore, we can conclude that x and y can be any real numbers.

Method 3: Graphical Method

We can also solve this system of equations by graphing the two equations on a coordinate plane.

The first equation is 2x + 2y = 12, which can be rewritten as y = -x + 6. The second equation is x + y = 6, which can be rewritten as y = -x + 6.

Both equations have the same slope and y-intercept, which means that they are parallel lines. Since the lines are parallel, they do not intersect, and therefore, there is no solution to the system of equations.

Conclusion

In this article, we have discussed three methods for solving a system of two linear equations with two variables. The substitution method, elimination method, and graphical method all lead to the same conclusion: there is no solution to the system of equations.

The system of equations is given as:

{ \left\{ \begin{array}{c} 2x + 2y = 12 \\ x + y = 6 \end{array} \right. \}

The solution to this system of equations is that there is no solution. The two equations are parallel lines, and therefore, they do not intersect.

Frequently Asked Questions

  • What is a system of equations? A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.
  • What are the methods for solving a system of equations? The methods for solving a system of equations include the substitution method, elimination method, and graphical method.
  • What is the solution to the system of equations given in this article? The solution to the system of equations is that there is no solution.

References

Further Reading

Introduction

In our previous article, we discussed the solution to a system of two linear equations with two variables. However, we received many questions from readers who were unsure about the concepts and methods involved in solving systems of equations. In this article, we will address some of the most frequently asked questions about systems of equations.

Q&A

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 methods for solving a system of equations?

A: The methods for solving a system of equations include the substitution method, elimination method, and graphical method.

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.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: What is the graphical method?

A: The graphical method involves graphing the two equations on a coordinate plane and finding the point of intersection.

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 and have two variables, the substitution or elimination method may be used. If the equations are non-linear or have more than two variables, the graphical method may be used.

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

A: A system of inequalities is a set of two or more inequalities that are solved simultaneously to find the values of the variables. The main difference between a system of equations and a system of inequalities is that inequalities involve a comparison between two expressions, whereas equations involve an equality between two expressions.

Q: Can a system of equations have no solution?

A: Yes, a system of equations can have no solution. This occurs when the two equations are parallel lines and do not intersect.

Q: Can a system of equations have an infinite number of solutions?

A: Yes, a system of equations can have an infinite number of solutions. This occurs when the two equations are identical and represent the same line.

Q: How do I graph a system of equations?

A: To graph a system of equations, first graph each equation separately on a coordinate plane. Then, find the point of intersection between the two graphs.

Q: What is the point of intersection?

A: The point of intersection is the point where the two graphs meet. This point represents the solution to the system of equations.

Q: Can a system of equations have multiple points of intersection?

A: No, a system of equations can have only one point of intersection. If the two graphs intersect at multiple points, it means that the system of equations has multiple solutions.

Conclusion

In this article, we have addressed some of the most frequently asked questions about systems of equations. We hope that this article has provided a clear understanding of the concepts and methods involved in solving systems of equations.

Frequently Asked Questions (FAQs)

  • What is a system of equations?
  • What are the methods for solving a system of equations?
  • What is the substitution method?
  • What is the elimination method?
  • What is the graphical method?
  • How do I know which method to use?
  • What is the difference between a system of equations and a system of inequalities?
  • Can a system of equations have no solution?
  • Can a system of equations have an infinite number of solutions?
  • How do I graph a system of equations?
  • What is the point of intersection?
  • Can a system of equations have multiple points of intersection?

References

Further Reading