Solve The System Of Equations:$\[ \begin{array}{c} 6x + 4y = 6 \\ 3x = -15 \\ \end{array} \\]

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.

The System of Equations

The given system of equations is:

{ \begin{array}{c} 6x + 4y = 6 \\ 3x = -15 \\ \end{array} \}

This system consists of two linear equations with two variables, x and y. The first equation is 6x + 4y = 6, and the second equation is 3x = -15.

Method 1: Substitution Method

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

Let's start by solving the second equation for x:

3x=−15x=−153x=−5{ 3x = -15 \\ x = \frac{-15}{3} \\ x = -5 \\ }

Now that we have the value of x, we can substitute it into the first equation:

6x+4y=66(−5)+4y=6−30+4y=64y=36y=364y=9{ 6x + 4y = 6 \\ 6(-5) + 4y = 6 \\ -30 + 4y = 6 \\ 4y = 36 \\ y = \frac{36}{4} \\ y = 9 \\ }

Therefore, the solution to the system of equations is x = -5 and y = 9.

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 of the variables.

Let's start by multiplying the second equation by 4 to make the coefficients of y in both equations the same:

4(3x)=4(−15)12x=−60{ 4(3x) = 4(-15) \\ 12x = -60 \\ }

Now we can add the two equations to eliminate the variable y:

(6x+4y)+(12x−60)=6+(−60)18x−60=−5418x=6x=618x=13{ (6x + 4y) + (12x - 60) = 6 + (-60) \\ 18x - 60 = -54 \\ 18x = 6 \\ x = \frac{6}{18} \\ x = \frac{1}{3} \\ }

Now that we have the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:

6x+4y=66(13)+4y=62+4y=64y=4y=44y=1{ 6x + 4y = 6 \\ 6(\frac{1}{3}) + 4y = 6 \\ 2 + 4y = 6 \\ 4y = 4 \\ y = \frac{4}{4} \\ y = 1 \\ }

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

Conclusion

In this article, we have solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. Both methods have given us the same solution, which is x = -5 and y = 9. This demonstrates that the solution to a system of linear equations is unique, and that different methods can be used to find the same solution.

Applications

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

  • Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.
  • Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.
  • Economics: Systems of linear equations are used to model economic systems, including supply and demand, and the behavior of markets.

Final Thoughts

Solving systems of linear equations is an important skill in mathematics, and it has many real-world applications. By using different methods, such as the substitution method and the elimination method, we can find the solution to a system of linear equations. Whether you are a student, a professional, or simply someone who is interested in mathematics, understanding how to solve systems of linear equations is an essential skill that can help you solve a wide range of problems.

References

  • [1] "Linear Algebra and Its Applications" by Gilbert Strang
  • [2] "Introduction to Linear Algebra" by Jim Hefferon
  • [3] "Linear Algebra: A Modern Introduction" by David Poole

Note: The references provided are a selection of popular textbooks on linear algebra. There are many other resources available, including online courses, videos, and tutorials.

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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 the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system.

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

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

  • Substitution Method: This method involves solving one of the equations for one variable and then substituting that expression into the other equation.
  • Elimination Method: This method involves adding or subtracting the equations to eliminate one of the variables.
  • Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
  • Matrix Method: This method involves using matrices to solve the system of equations.

Q: What is the difference between the substitution method and the elimination method?

A: The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I know which method to use?

A: The choice of method depends on the specific system of equations and the variables involved. If the system has two variables and two equations, the substitution method and the elimination method are often the most effective. If the system has more than two variables or equations, the matrix method may be more efficient.

Q: What if I have a system of linear equations with more than two variables?

A: If you have a system of linear equations with more than two variables, you can use the matrix method to solve the system. This involves representing the system as a matrix and using row operations to find the solution.

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

A: Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations, including the substitution method and the elimination method.

Q: What if I have a system of linear equations with no solution?

A: If you have a system of linear equations with no solution, it means that the equations are inconsistent. This can happen if the equations are contradictory or if the variables are not related in a way that satisfies the equations.

Q: What if I have a system of linear equations with infinitely many solutions?

A: If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent. This can happen if the equations are identical or if one equation is a multiple of the other.

Q: How do I know if a system of linear equations has a unique solution, no solution, or infinitely many solutions?

A: To determine the nature of the solution, you can use the following criteria:

  • Unique Solution: If the system has a unique solution, it means that the equations are consistent and the variables are related in a way that satisfies the equations.
  • No Solution: If the system has no solution, it means that the equations are inconsistent and the variables are not related in a way that satisfies the equations.
  • Infinitely Many Solutions: If the system has infinitely many solutions, it means that the equations are dependent and the variables are related in a way that satisfies the equations.

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

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

  • Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.
  • Computer Science: Systems of linear equations are used in computer graphics, game development, and machine learning.
  • Economics: Systems of linear equations are used to model economic systems, including supply and demand, and the behavior of markets.

Q: How can I practice solving systems of linear equations?

A: You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also use online tools and calculators to help you solve systems of linear equations.

Q: What are some common mistakes to avoid when solving systems of linear equations?

A: Some common mistakes to avoid when solving systems of linear equations include:

  • Not checking for consistency: Make sure to check if the equations are consistent before solving the system.
  • Not using the correct method: Choose the correct method for solving the system, such as the substitution method or the elimination method.
  • Not checking for dependent or inconsistent equations: Make sure to check if the equations are dependent or inconsistent before solving the system.

By following these tips and practicing regularly, you can become proficient in solving systems of linear equations and apply this skill to a wide range of real-world problems.