Solve The System Of Equations:1. \[$ 2x + Y = -6 \$\]2. The Solution Is \[$\square\$\].

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Introduction

Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the solution.

What are Systems of Linear Equations?

A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. For example:

  • 2x + y = -6
  • x - 2y = 4

Why is Solving Systems of Linear Equations Important?

Solving systems of linear equations is essential in various fields, including:

  • Science and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects, electrical circuits, and population growth.
  • Computer Science: Linear equations are used in computer graphics, game development, and machine learning.
  • Economics: Systems of linear equations are used to model economic systems, such as supply and demand, and resource allocation.

Methods for Solving Systems of Linear Equations

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

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

Solving the System of Equations

Let's solve the system of equations:

  1. 2x + y = -6
  2. x - 2y = 4

Step 1: Write Down the Equations

We have two equations:

  1. 2x + y = -6
  2. x - 2y = 4

Step 2: Solve One Equation for One Variable

Let's solve the second equation for x:

x = 4 + 2y

Step 3: Substitute the Expression into the Other Equation

Substitute the expression for x into the first equation:

2(4 + 2y) + y = -6

Step 4: Simplify the Equation

Expand and simplify the equation:

8 + 4y + y = -6

Combine like terms:

5y + 8 = -6

Step 5: Solve for y

Subtract 8 from both sides:

5y = -14

Divide both sides by 5:

y = -14/5

Step 6: Find the Value of x

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

x = 4 + 2(-14/5)

Simplify the expression:

x = 4 - 28/5

x = (20 - 28)/5

x = -8/5

Step 7: Write the Solution

The solution to the system of equations is:

x = -8/5 y = -14/5

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics. In this article, we used the method of substitution and elimination to solve a system of two linear equations with two variables. We found the values of x and y that satisfy both equations simultaneously. Solving systems of linear equations is essential in various fields, including science, engineering, computer science, and economics.

Final Answer

The final answer is: (−85,−145)\boxed{\left(-\frac{8}{5}, -\frac{14}{5}\right)}

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Introduction

Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we provided a step-by-step guide on how to solve a system of two linear equations with two variables. In this article, we will answer some frequently asked questions (FAQs) related to solving systems of linear equations.

Q&A

Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: Why is solving systems of linear equations important?

Solving systems of linear equations is essential in various fields, including science, engineering, computer science, and economics. It helps us model real-world problems, make predictions, and optimize solutions.

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

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

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

Q: How do I choose the right method for solving a system of linear equations?

The choice of method depends on the type of equations and the variables involved. If the equations are simple and have a clear solution, the substitution method may be the best choice. If the equations are complex or have multiple solutions, the elimination method may be more suitable.

Q: What if I have a system of linear equations with three or more variables?

If you have a system of linear equations with three or more variables, you can use the same methods as before, but with additional steps. You may need to use the elimination method to reduce the number of variables, and then use the substitution method to solve for the remaining variables.

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

Yes, you can use technology, such as calculators or computer software, to solve systems of linear equations. Many calculators and software programs have built-in functions for solving systems of linear equations, making it easier to find the solution.

Q: What if I have a system of linear equations with no solution or infinitely many solutions?

If you have a system of linear equations with no solution or infinitely many solutions, it means that the equations are inconsistent or dependent. In this case, you can use the elimination method to determine the nature of the solution.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics, and it has many practical applications in various fields. By understanding the different methods for solving systems of linear equations, you can tackle complex problems and make predictions with confidence.

Final Tips

  • Practice solving systems of linear equations with different methods to become proficient.
  • Use technology, such as calculators or computer software, to check your solutions.
  • Be careful when solving systems of linear equations with no solution or infinitely many solutions.

Common Mistakes to Avoid

  • Not checking the solution for consistency.
  • Not using the correct method for solving the system of linear equations.
  • Not considering the possibility of no solution or infinitely many solutions.

Additional Resources

  • Online resources, such as Khan Academy or MIT OpenCourseWare, offer video lectures and practice problems for solving systems of linear equations.
  • Textbooks, such as "Linear Algebra and Its Applications" by Gilbert Strang, provide in-depth explanations and examples.
  • Online communities, such as Reddit's r/learnmath, offer support and guidance for students learning mathematics.