Solve The System By Elimination.${ \begin{array}{l} \left{ \begin{array}{l} 2x + Y = \square \ -x - Y = \square \end{array} \right. \ (x, Y) = (\square) \end{array} }$

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Introduction

In this article, we will explore the method of elimination to solve a system of linear equations. The method of elimination is a powerful technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the equations are integers or fractions.

What is the Method of Elimination?

The method of elimination is a step-by-step process that involves adding or subtracting equations to eliminate one of the variables. The goal is to create an equation with only one variable, which can then be solved for that variable. Once the value of one variable is known, it can be substituted into one of the original equations to solve for the other variable.

Step 1: Write Down the Equations

The first step in solving a system of linear equations by elimination is to write down the equations. In this case, we have two equations:

  • 2x + y = 7
  • -x - y = -3

Step 2: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same. In this case, we can multiply the first equation by 1 and the second equation by 2.

  • 2x + y = 7
  • -2x - 2y = -6

Step 3: Add or Subtract the Equations

Now that we have the equations with the same coefficients for the variable to be eliminated, we can add or subtract them to eliminate that variable. In this case, we can add the two equations to eliminate the variable x.

  • (2x - 2x) + (y - 2y) = 7 - 6
  • -y = 1

Step 4: Solve for the Variable

Now that we have an equation with only one variable, we can solve for that variable. In this case, we can solve for y.

  • -y = 1
  • y = -1

Step 5: Substitute the Value of the Variable into One of the Original Equations

Now that we have the value of one variable, we can substitute it into one of the original equations to solve for the other variable. In this case, we can substitute y = -1 into the first equation.

  • 2x + (-1) = 7
  • 2x = 8
  • x = 4

Conclusion

In this article, we have explored the method of elimination to solve a system of linear equations. We have seen how to write down the equations, multiply them by necessary multiples, add or subtract them to eliminate one of the variables, solve for the variable, and substitute the value of the variable into one of the original equations to solve for the other variable. The method of elimination is a powerful technique used to solve systems of linear equations and is particularly useful when the coefficients of the variables in the equations are integers or fractions.

Example 2: Solve the System by Elimination

Let's consider another example of a system of linear equations.

  • x + 2y = 6
  • 3x - 2y = 2

To solve this system by elimination, we can multiply the first equation by 2 and the second equation by 1.

  • 2x + 4y = 12
  • 3x - 2y = 2

Now, we can add the two equations to eliminate the variable y.

  • (2x + 3x) + (4y - 2y) = 12 + 2
  • 5x + 2y = 14

Now, we can subtract the second equation from the first equation to eliminate the variable y.

  • (5x - 3x) + (2y + 2y) = 14 - 2
  • 2x + 4y = 12

Now, we can solve for the variable x.

  • 2x + 4y = 12
  • 2x = 12 - 4y
  • x = (12 - 4y) / 2

Now, we can substitute the value of x into one of the original equations to solve for the variable y.

  • x + 2y = 6
  • (12 - 4y) / 2 + 2y = 6
  • 12 - 4y + 4y = 12
  • 12 = 12

This is a contradiction, which means that the system of linear equations has no solution.

Conclusion

In this article, we have explored the method of elimination to solve a system of linear equations. We have seen how to write down the equations, multiply them by necessary multiples, add or subtract them to eliminate one of the variables, solve for the variable, and substitute the value of the variable into one of the original equations to solve for the other variable. The method of elimination is a powerful technique used to solve systems of linear equations and is particularly useful when the coefficients of the variables in the equations are integers or fractions.

Applications of the Method of Elimination

The method of elimination has many applications in mathematics and other fields. Some of the applications of the method of elimination include:

  • Solving systems of linear equations
  • Finding the intersection of two lines
  • Finding the equation of a line passing through two points
  • Solving systems of linear inequalities
  • Finding the solution set of a system of linear equations

Conclusion

In conclusion, the method of elimination is a powerful technique used to solve systems of linear equations. It involves writing down the equations, multiplying them by necessary multiples, adding or subtracting them to eliminate one of the variables, solving for the variable, and substituting the value of the variable into one of the original equations to solve for the other variable. The method of elimination has many applications in mathematics and other fields, and is particularly useful when the coefficients of the variables in the equations are integers or fractions.

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

Further Reading

For further reading on the method of elimination, we recommend the following resources:

  • "Linear Algebra and Its Applications" by Gilbert Strang
  • "Introduction to Linear Algebra" by Jim Hefferon
  • "Linear Algebra: A Modern Introduction" by David Poole

We hope that this article has provided a comprehensive overview of the method of elimination and its applications. If you have any further questions or need additional clarification, please don't hesitate to ask.

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Q1: What is the method of elimination in solving systems of linear equations?

A1: The method of elimination is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the equations are integers or fractions.

Q2: How do I choose which variable to eliminate first?

A2: To choose which variable to eliminate first, you need to look at the coefficients of the variables in the equations. You want to eliminate the variable that has the smallest coefficient. This is because the smaller coefficient will be easier to eliminate.

Q3: What if the coefficients of the variables are the same?

A3: If the coefficients of the variables are the same, you can multiply one of the equations by a number that will make the coefficients different. For example, if you have two equations with the same coefficient for x, you can multiply one of the equations by 2 to make the coefficients different.

Q4: How do I add or subtract the equations to eliminate a variable?

A4: To add or subtract the equations to eliminate a variable, you need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same. Then, you can add or subtract the equations to eliminate the variable.

Q5: What if I get a contradiction when I add or subtract the equations?

A5: If you get a contradiction when you add or subtract the equations, it means that the system of linear equations has no solution. This is because the equations are inconsistent and cannot be solved simultaneously.

Q6: Can I use the method of elimination to solve systems of linear inequalities?

A6: No, the method of elimination is only used to solve systems of linear equations, not systems of linear inequalities. Systems of linear inequalities are solved using a different method, such as the graphical method or the simplex method.

Q7: What are some common mistakes to avoid when using the method of elimination?

A7: Some common mistakes to avoid when using the method of elimination include:

  • Not multiplying the equations by necessary multiples to make the coefficients of the variable to be eliminated the same.
  • Not adding or subtracting the equations correctly to eliminate the variable.
  • Not checking for contradictions when adding or subtracting the equations.
  • Not solving for the variable correctly after eliminating it.

Q8: Can I use the method of elimination to solve systems of linear equations with fractions?

A8: Yes, you can use the method of elimination to solve systems of linear equations with fractions. However, you need to be careful when multiplying the equations by necessary multiples to make the coefficients of the variable to be eliminated the same.

Q9: What are some real-world applications of the method of elimination?

A9: Some real-world applications of the method of elimination include:

  • Solving systems of linear equations in physics and engineering to find the values of variables such as distance, time, and velocity.
  • Solving systems of linear equations in economics to find the values of variables such as supply and demand.
  • Solving systems of linear equations in computer science to find the values of variables such as memory and processing power.

Q10: Can I use the method of elimination to solve systems of linear equations with more than two variables?

A10: Yes, you can use the method of elimination to solve systems of linear equations with more than two variables. However, you need to be careful when adding or subtracting the equations to eliminate the variables.

Conclusion

In conclusion, the method of elimination is a powerful technique used to solve systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, solving for the variable, and substituting the value of the variable into one of the original equations to solve for the other variable. By following the steps outlined in this article, you can use the method of elimination to solve systems of linear equations and apply it to real-world 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

Further Reading

For further reading on the method of elimination, we recommend the following resources:

  • "Linear Algebra and Its Applications" by Gilbert Strang
  • "Introduction to Linear Algebra" by Jim Hefferon
  • "Linear Algebra: A Modern Introduction" by David Poole

We hope that this article has provided a comprehensive overview of the method of elimination and its applications. If you have any further questions or need additional clarification, please don't hesitate to ask.