Solve The System Of Equations By Elimination:$\[ \begin{array}{l} 5x + Y = 9 \\ 10x - 7y = -18 \end{array} \\]A. \[$(-1, -4)\$\] B. \[$(4, 1)\$\] C. \[$(1, 4)\$\] D. \[$(-4, -1)\$\]

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Introduction

In this article, we will learn how to solve a system of linear equations using the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables and then solving for the other variable. We will use a step-by-step approach to solve the system of equations and find the solution.

What is a System of Equations?

A system of equations is a set of two or more equations that contain the same variables. In this case, we have two equations with two variables, x and y. The system of equations is:

{ \begin{array}{l} 5x + y = 9 \\ 10x - 7y = -18 \end{array} \}

The Elimination Method

The elimination method involves adding or subtracting the equations to eliminate one of the variables. To do this, we need to multiply the equations by necessary multiples such that the coefficients of either x or y will be the same in both equations but with opposite signs.

Step 1: Multiply the Equations

We will multiply the first equation by 7 and the second equation by 1. This will give us:

{ \begin{array}{l} 35x + 7y = 63 \\ 10x - 7y = -18 \end{array} \}

Step 2: Add the Equations

Now, we will add the two equations to eliminate the variable y. This will give us:

{ 45x = 45 \}

Step 3: Solve for x

Now, we will solve for x by dividing both sides of the equation by 45. This will give us:

{ x = 1 \}

Step 4: Substitute x into One of the Original Equations

Now, we will substitute x into one of the original equations to solve for y. We will use the first equation:

{ 5x + y = 9 \}

Substituting x = 1, we get:

{ 5(1) + y = 9 \}

Simplifying, we get:

{ 5 + y = 9 \}

Subtracting 5 from both sides, we get:

{ y = 4 \}

Conclusion

Therefore, the solution to the system of equations is x = 1 and y = 4.

Answer

The correct answer is:

{ (1, 4) \}

This is option C.

Why is the Elimination Method Important?

The elimination method is an important technique in solving systems of linear equations. It is a simple and efficient method that can be used to solve systems of equations with two or more variables. The elimination method is also a useful tool in many real-world applications, such as physics, engineering, and economics.

Real-World Applications of the Elimination Method

The elimination method has many real-world applications. Some examples include:

  • Solving systems of equations in physics to find the position and velocity of an object.
  • Solving systems of equations in engineering to design and optimize systems.
  • Solving systems of equations in economics to model and analyze economic systems.

Conclusion

In conclusion, the elimination method is a powerful technique for solving systems of linear equations. It is a simple and efficient method that can be used to solve systems of equations with two or more variables. The elimination method is also a useful tool in many real-world applications, such as physics, engineering, and economics.

Final Thoughts

The elimination method is an important technique in mathematics and has many real-world applications. It is a simple and efficient method that can be used to solve systems of equations with two or more variables. By understanding the elimination method, you can solve systems of equations and apply the technique to many real-world problems.

References

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

Additional Resources

  • Khan Academy: Linear Equations and Inequalities
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram MathWorld: Linear Algebra
    Solve the System of Equations by Elimination: Q&A =====================================================

Introduction

In our previous article, we learned how to solve a system of linear equations using the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables and then solving for the other variable. In this article, we will answer some frequently asked questions about the elimination method and provide additional examples to help you understand the concept better.

Q: What is the elimination method?

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

Q: How do I choose which variable to eliminate?

A: To choose which variable to eliminate, you need to look at the coefficients of the variables in both equations. If the coefficients of one variable are the same in both equations but with opposite signs, you can eliminate that variable by adding or subtracting the equations.

Q: What if the coefficients of the variables are not the same in both equations?

A: If the coefficients of the variables are not the same in both equations, you need to multiply one or both of the equations by necessary multiples such that the coefficients of either x or y will be the same in both equations but with opposite signs.

Q: Can I use the elimination method to solve systems of equations with three or more variables?

A: Yes, you can use the elimination method to solve systems of equations with three or more variables. However, it may be more complicated and require more steps.

Q: What are some common mistakes to avoid when using the elimination method?

A: Some common mistakes to avoid when using the elimination method include:

  • Not checking if the coefficients of the variables are the same in both equations before eliminating a variable.
  • Not multiplying one or both of the equations by necessary multiples to make the coefficients of either x or y the same in both equations but with opposite signs.
  • Not solving for the remaining variable after eliminating one variable.

Q: Can I use the elimination method to solve systems of equations with fractions or decimals?

A: Yes, you can use the elimination method to solve systems of equations with fractions or decimals. However, you need to follow the same steps as you would with integers.

Q: What are some real-world applications of the elimination method?

A: Some real-world applications of the elimination method include:

  • Solving systems of equations in physics to find the position and velocity of an object.
  • Solving systems of equations in engineering to design and optimize systems.
  • Solving systems of equations in economics to model and analyze economic systems.

Q: How do I know if the elimination method is the best method to use to solve a system of equations?

A: To determine if the elimination method is the best method to use to solve a system of equations, you need to consider the following factors:

  • The number of variables in the system of equations.
  • The complexity of the system of equations.
  • The type of solution you are looking for (e.g., exact or approximate).

Conclusion

In conclusion, the elimination method is a powerful technique for solving systems of linear equations. It is a simple and efficient method that can be used to solve systems of equations with two or more variables. By understanding the elimination method, you can solve systems of equations and apply the technique to many real-world problems.

Final Thoughts

The elimination method is an important technique in mathematics and has many real-world applications. It is a simple and efficient method that can be used to solve systems of equations with two or more variables. By understanding the elimination method, you can solve systems of equations and apply the technique to many real-world problems.

References

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

Additional Resources

  • Khan Academy: Linear Equations and Inequalities
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram MathWorld: Linear Algebra

Practice Problems

  1. Solve the system of equations using the elimination method:

{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \}

  1. Solve the system of equations using the elimination method:

{ \begin{array}{l} x + 2y = 6 \\ 3x - 4y = -2 \end{array} \}

  1. Solve the system of equations using the elimination method:

{ \begin{array}{l} x - 3y = -2 \\ 2x + 5y = 11 \end{array} \}

Answer Key

  1. (x, y) = (4, 1)
  2. (x, y) = (2, 2)
  3. (x, y) = (1, 2)

Conclusion

In conclusion, the elimination method is a powerful technique for solving systems of linear equations. It is a simple and efficient method that can be used to solve systems of equations with two or more variables. By understanding the elimination method, you can solve systems of equations and apply the technique to many real-world problems.