What Is The Solution To This System Of Linear Equations?$\[ \begin{array}{l} y - 4x = 7 \\ 2y + 4x = 2 \end{array} \\]A. (3, 1) B. (1, 3) C. (3, -1) D. (-1, 3)

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Introduction to Systems of Linear Equations

Systems of linear equations are a fundamental concept in mathematics, particularly in algebra and geometry. They consist of two or more linear equations that involve variables, and the goal is to find the values of these variables that satisfy all the equations simultaneously. In this article, we will focus on solving a specific system of linear equations and explore the different methods used to find the solution.

The System of Linear Equations

The given system of linear equations is:

{ \begin{array}{l} y - 4x = 7 \\ 2y + 4x = 2 \end{array} \}

This system consists of two linear equations with two variables, x and y. The first equation is y - 4x = 7, and the second equation is 2y + 4x = 2.

Method 1: Substitution Method

One of the methods used to solve systems of linear equations is the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Let's start by solving the first equation for y:

y = 4x + 7

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

2(4x + 7) + 4x = 2

Expand and simplify the equation:

8x + 14 + 4x = 2

Combine like terms:

12x + 14 = 2

Subtract 14 from both sides:

12x = -12

Divide both sides by 12:

x = -1

Now that we have found the value of x, substitute it back into the expression for y:

y = 4(-1) + 7

y = -4 + 7

y = 3

Therefore, the solution to the system of linear equations is x = -1 and y = 3.

Method 2: Elimination Method

Another method used to solve systems of linear equations is the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables.

Let's start by multiplying the first equation by 2 to make the coefficients of y in both equations the same:

2(y - 4x) = 2(7)

2y - 8x = 14

Now, add the two equations to eliminate the variable y:

(2y - 8x) + (2y + 4x) = 14 + 2

Combine like terms:

4y - 4x = 16

Divide both sides by 4:

y - x = 4

Now, substitute this expression for y into one of the original equations. Let's use the first equation:

y - 4x = 7

Substitute y = x + 4:

(x + 4) - 4x = 7

Combine like terms:

-x + 4 = 7

Subtract 4 from both sides:

-x = 3

Divide both sides by -1:

x = -3

Now that we have found the value of x, substitute it back into the expression for y:

y = x + 4

y = -3 + 4

y = 1

Therefore, the solution to the system of linear equations is x = -3 and y = 1.

Conclusion

In this article, we have explored two methods used to solve systems of linear equations: the substitution method and the elimination method. We have applied these methods to a specific system of linear equations and found the solution to be x = -1 and y = 3 using the substitution method, and x = -3 and y = 1 using the elimination method. These methods are essential tools in mathematics and are widely used in various fields, including science, engineering, and economics.

Frequently Asked Questions

  • What is a system of linear equations? A system of linear equations is a set of two or more linear equations that involve variables, and the goal is to find the values of these variables that satisfy all the equations simultaneously.
  • What are the two methods used to solve systems of linear equations? The two methods used to solve systems of linear equations are the substitution method and the elimination method.
  • How do I choose which method to use? The choice of method depends on the specific system of linear equations and the variables involved. The substitution method is often used when one equation is easily solvable for one variable, while the elimination method is often used when the coefficients of the variables are the same in both equations.

References

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

Introduction

Systems of linear equations are a fundamental concept in mathematics, particularly in algebra and geometry. They consist of two or more linear equations that involve variables, and the goal is to find the values of these variables that satisfy all the equations simultaneously. In this article, we will answer some of the most frequently asked questions about systems of linear equations.

Q&A

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve variables, and the goal is to find the values of these variables that satisfy all the equations simultaneously.

Q: What are the two methods used to solve systems of linear equations?

A: The two methods used to solve systems of linear equations are the substitution method and the elimination method.

Q: How do I choose which method to use?

A: The choice of method depends on the specific system of linear equations and the variables involved. The substitution method is often used when one equation is easily solvable for one variable, while the elimination method is often used when the coefficients of the variables are the same in both equations.

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 of the variables.

Q: How do I know if a system of linear equations has a solution?

A: A system of linear equations has a solution if the two equations are consistent, meaning that they do not contradict each other.

Q: How do I know if a system of linear equations has no solution?

A: A system of linear equations has no solution if the two equations are inconsistent, meaning that they contradict each other.

Q: How do I know if a system of linear equations has infinitely many solutions?

A: A system of linear equations has infinitely many solutions if the two equations are equivalent, meaning that they represent the same line.

Q: Can I use a graphing calculator to solve systems of linear equations?

A: Yes, you can use a graphing calculator to solve systems of linear equations by graphing the two equations and finding the point of intersection.

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

A: Yes, you can use a computer program such as MATLAB or Python to solve systems of linear equations using numerical methods.

Q: What are some real-world applications of systems of linear equations?

A: Systems of linear equations have many real-world applications, including:

  • Physics: to describe the motion of objects
  • Engineering: to design and optimize systems
  • Economics: to model economic systems
  • Computer Science: to solve problems in computer graphics and game development

Conclusion

In this article, we have answered some of the most frequently asked questions about systems of linear equations. We have covered the basics of systems of linear equations, including the two methods used to solve them, and some real-world applications. We hope that this article has been helpful in answering your questions and providing a better understanding of systems of linear equations.

Frequently Asked Questions (FAQs)

  • What is a system of linear equations?
  • What are the two methods used to solve systems of linear equations?
  • How do I choose which method to use?
  • What is the substitution method?
  • What is the elimination method?
  • How do I know if a system of linear equations has a solution?
  • How do I know if a system of linear equations has no solution?
  • How do I know if a system of linear equations has infinitely many solutions?
  • Can I use a graphing calculator to solve systems of linear equations?
  • Can I use a computer program to solve systems of linear equations?
  • What are some real-world applications of systems of linear equations?

References

  • [1] "Systems of Linear Equations" by Math Open Reference
  • [2] "Solving Systems of Linear Equations" by Khan Academy
  • [3] "Systems of Linear Equations" by Wolfram MathWorld
  • [4] "Linear Algebra and Its Applications" by Gilbert Strang
  • [5] "Introduction to Linear Algebra" by Jim Hefferon