Solve The System Of Equations Using Elimination.${ \begin{array}{l} -2x + 3y = 13 \ x + Y = 11 \end{array} }$Choose The Correct Solution:A. $(-5, 1)$ B. $(-3, 3)$ C. $(3, 8)$ D. $(4, 7)$

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. There are several methods to solve a system of equations, including substitution, elimination, and graphing. In this article, we will focus on the elimination method to solve a system of linear equations.

What is the Elimination Method?

The elimination method is a technique used to solve a system of linear equations by adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of the variables in the two equations are additive inverses of each other.

Step-by-Step Guide to Solving a System of Equations Using Elimination

To solve a system of equations using elimination, follow these steps:

Step 1: Write Down the System of Equations

Write down the system of equations that you want to solve. In this case, we have two equations:

{ \begin{array}{l} -2x + 3y = 13 \\ x + y = 11 \end{array} \}

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.

{ \begin{array}{l} -2x + 3y = 13 \\ 2x + 2y = 22 \end{array} \}

Step 3: Add or Subtract the Equations

Now, we can add or subtract the equations to eliminate one of the variables. In this case, we can add the two equations to eliminate the variable x.

{ -2x + 3y = 13 \\ 2x + 2y = 22 \end{array} \}

Adding the two equations, we get:

{ 5y = 35 \}

Step 4: Solve for the Variable

Now, we can solve for the variable y by dividing both sides of the equation by 5.

{ y = 7 \}

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

Now that we have the value of y, we can substitute it into one of the original equations to solve for the other variable. Let's substitute y = 7 into the second original equation.

{ x + 7 = 11 \}

Step 6: Solve for the Other Variable

Now, we can solve for the other variable x by subtracting 7 from both sides of the equation.

{ x = 4 \}

Conclusion

In this article, we have learned how to solve a system of equations using the elimination method. We have followed the steps to eliminate one of the variables, solve for the other variable, and find the solution to the system of equations. The solution to the system of equations is (4, 7).

Answer

The correct solution is:

D. (4, 7)

Discussion

This problem is a great example of how to use the elimination method to solve a system of equations. The elimination method is a powerful tool for solving systems of equations, and it is essential to understand how to use it correctly. In this case, we were able to eliminate the variable x by adding the two equations, and then we were able to solve for the other variable y by substituting the value of x into one of the original equations.

Tips and Variations

Here are some tips and variations to keep in mind when using the elimination method:

• Make sure to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same.
• Be careful when adding or subtracting the equations, as this can lead to errors.
• If the coefficients of the variables are not additive inverses of each other, you may need to use a different method to solve the system of equations.
• You can also use the elimination method to solve systems of equations with more than two variables.

Real-World Applications

The elimination method has many real-world applications, including:

• Solving systems of equations in physics and engineering
• Finding the intersection of two lines in geometry
• Solving systems of equations in economics and finance

Conclusion

Introduction

In our previous article, we learned how to solve a system of equations using the elimination method. In this article, we will answer some frequently asked questions about the elimination method and provide additional examples to help you practice.

Q: What is the elimination method?

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

Q: How do I know which variable to eliminate?

A: To eliminate a variable, you need to make the coefficients of that variable the same in both equations. You can do this by multiplying one or both of the equations by necessary multiples.

Q: What if the coefficients of the variables are not additive inverses of each other?

A: If the coefficients of the variables are not additive inverses of each other, you may need to use a different method to solve the system of equations, such as substitution or graphing.

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

A: Yes, you can use the elimination method to solve systems of equations with more than two variables. However, you will need to eliminate variables one at a time, using the same steps as before.

Q: How do I know if I have found the correct solution?

A: To check if you have found the correct solution, substitute the values of the variables back into the original equations and make sure they are true.

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

A: If you get a system of equations with no solution, it means that the equations are inconsistent and there is no value of the variables that can satisfy both equations.

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

A: If you get a system of equations with infinitely many solutions, it means that the equations are dependent and there are many values of the variables that can satisfy both equations.

Examples

Here are some examples to help you practice using the elimination method:

Example 1

Solve the system of equations:

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

Solution

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

{ \begin{array}{l} 2x + 3y = 7 \\ 2x - 4y = -6 \end{array} \}

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

{ 7y = 1 \}

Solving for y, we get:

{ y = \frac{1}{7} \}

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

{ 2x + 3\left(\frac{1}{7}\right) = 7 \}

Solving for x, we get:

{ x = \frac{40}{7} \}

Example 2

Solve the system of equations:

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

Solution

To solve this system of equations, we can multiply the first equation by 3 and the second equation by 1.

{ \begin{array}{l} 3x + 6y = 12 \\ 3x - 2y = 2 \end{array} \}

Now, we can subtract the two equations to eliminate the variable x.

{ 8y = 10 \}

Solving for y, we get:

{ y = \frac{5}{4} \}

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

{ x + 2\left(\frac{5}{4}\right) = 4 \}

Solving for x, we get:

{ x = \frac{8}{4} \}

Conclusion

In this article, we have answered some frequently asked questions about the elimination method and provided additional examples to help you practice. Remember to multiply the equations by necessary multiples, add or subtract the equations, and solve for the variables. With practice and patience, you will become proficient in using the elimination method to solve systems of equations.

Tips and Variations

Here are some tips and variations to keep in mind when using the elimination method:

• Make sure to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same.
• Be careful when adding or subtracting the equations, as this can lead to errors.
• If the coefficients of the variables are not additive inverses of each other, you may need to use a different method to solve the system of equations.
• You can also use the elimination method to solve systems of equations with more than two variables.

Real-World Applications

The elimination method has many real-world applications, including:

• Solving systems of equations in physics and engineering
• Finding the intersection of two lines in geometry
• Solving systems of equations in economics and finance

Conclusion

In conclusion, the elimination method is a powerful tool for solving systems of equations. By following the steps outlined in this article, you can use the elimination method to solve systems of equations and find the solution. Remember to multiply the equations by necessary multiples, add or subtract the equations, and solve for the variables. With practice and patience, you will become proficient in using the elimination method to solve systems of equations.