To Solve The System Of Linear Equations 3 X − 2 Y = 4 3x - 2y = 4 3 X − 2 Y = 4 And 9 X − 6 Y = 12 9x - 6y = 12 9 X − 6 Y = 12 Using The Linear Combination Method, Henry Decided To Multiply The First Equation By -3 And Then Add The Two Equations Together To Eliminate The

Mar 11, 2025 by ADMIN 272 views

**To Solve the System of Linear Equations Using Linear Combination Method**

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Introduction

In mathematics, solving a system of linear equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. One of the methods used to solve such systems is the linear combination method, which involves combining two or more equations to eliminate one of the variables. In this article, we will explore how to use the linear combination method to solve a system of linear equations, using the example of the system $3x - 2y = 4$ and $9x - 6y = 12$ .

Understanding the Linear Combination Method

The linear combination method involves multiplying one or more equations by a constant and then adding the resulting equations together to eliminate one of the variables. This method is based on the principle that if two equations are added together, the coefficients of the variable to be eliminated will cancel each other out, leaving only the constant terms. The resulting equation will have only one variable, which can be solved to find its value.

Applying the Linear Combination Method to the Given System

Let's apply the linear combination method to the system of linear equations $3x - 2y = 4$ and $9x - 6y = 12$ . To eliminate the variable $y$ , we need to multiply the first equation by a constant that will make the coefficient of $y$ in both equations equal. In this case, we can multiply the first equation by $-3$ to make the coefficient of $y$ in both equations equal to $6$ .

Multiplying the First Equation by -3

-3(3x - 2y = 4)
-9x + 6y = -12

Adding the Two Equations Together

Now that we have multiplied the first equation by $-3$ , we can add the two equations together to eliminate the variable $y$ .

(9x - 6y = 12)
-9x + 6y = -12
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0x = 0

As we can see, the variable $y$ has been eliminated, and we are left with an equation that has only one variable, $x$ . However, this equation is not very useful, as it does not provide any information about the value of $x$ . To find the value of $x$ , we need to go back to one of the original equations and substitute the value of $y$ that we found earlier.

Finding the Value of x

Let's go back to the first equation and substitute the value of $y$ that we found earlier.

3x - 2y = 4
3x - 2(0) = 4
3x = 4
x = 4/3

Therefore, the value of $x$ is $4/3$ .

Conclusion

In this article, we have explored how to use the linear combination method to solve a system of linear equations. We have applied this method to the system $3x - 2y = 4$ and $9x - 6y = 12$ and found the values of the variables $x$ and $y$ . The linear combination method is a powerful tool for solving systems of linear equations, and it can be used to solve a wide range of problems in mathematics and other fields.

Example Problems

Problem 1

Solve the system of linear equations $2x + 3y = 5$ and $4x + 6y = 10$ using the linear combination method.

Problem 2

Solve the system of linear equations $x - 2y = 3$ and $2x - 4y = 6$ using the linear combination method.

Problem 3

Solve the system of linear equations $3x + 2y = 7$ and $6x + 4y = 14$ using the linear combination method.

Tips and Tricks

When using the linear combination method, make sure to multiply the equations by the correct constants to eliminate the variable.
When adding the equations together, make sure to add the constant terms correctly.
When substituting the value of one variable into an equation, make sure to substitute it correctly.

Frequently Asked Questions

Q: What is the linear combination method?

A: The linear combination method is a method used to solve systems of linear equations by combining two or more equations to eliminate one of the variables.

Q: How do I apply the linear combination method to a system of linear equations?

A: To apply the linear combination method, multiply one or more equations by a constant and then add the resulting equations together to eliminate one of the variables.

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

A: Some common mistakes to avoid when using the linear combination method include multiplying the equations by the wrong constants, adding the constant terms incorrectly, and substituting the value of one variable into an equation incorrectly.

Conclusion

In conclusion, the linear combination method is a powerful tool for solving systems of linear equations. By following the steps outlined in this article, you can use the linear combination method to solve a wide range of problems in mathematics and other fields. Remember to be careful when applying the linear combination method, and avoid common mistakes such as multiplying the equations by the wrong constants and adding the constant terms incorrectly.

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Q: What is the linear combination method?

A: The linear combination method is a method used to solve systems of linear equations by combining two or more equations to eliminate one of the variables. This method involves multiplying one or more equations by a constant and then adding the resulting equations together to eliminate one of the variables.

Q: How do I apply the linear combination method to a system of linear equations?

A: To apply the linear combination method, follow these steps:

Identify the system of linear equations that you want to solve.
Determine which variable you want to eliminate.
Multiply one or more equations by a constant to make the coefficient of the variable you want to eliminate equal in both equations.
Add the resulting equations together to eliminate the variable.
Solve the resulting equation for the remaining variable.

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

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

Multiplying the equations by the wrong constants.
Adding the constant terms incorrectly.
Substituting the value of one variable into an equation incorrectly.
Not checking the resulting equation for any errors or inconsistencies.

Q: Can I use the linear combination method to solve a system of linear equations with more than two variables?

A: Yes, you can use the linear combination method to solve a system of linear equations with more than two variables. However, you will need to eliminate one variable at a time, using the linear combination method to eliminate each variable in turn.

Q: How do I know which variable to eliminate first?

A: To determine which variable to eliminate first, look at the coefficients of the variables in the equations. The variable with the smallest coefficient is usually the best one to eliminate first.

Q: Can I use the linear combination method to solve a system of linear equations with fractions?

A: Yes, you can use the linear combination method to solve a system of linear equations with fractions. However, you will need to multiply the equations by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: How do I check my work when using the linear combination method?

A: To check your work when using the linear combination method, follow these steps:

Verify that the resulting equation is true for all values of the variables.
Check that the solution satisfies all of the original equations.
Use a graphing calculator or computer software to check the solution.

Q: Can I use the linear combination method to solve a system of linear equations with no solution?

A: Yes, you can use the linear combination method to solve a system of linear equations with no solution. However, if the system has no solution, the resulting equation will be a contradiction, and you will not be able to find a solution.

Q: How do I know if a system of linear equations has no solution?

A: To determine if a system of linear equations has no solution, follow these steps:

Check if the resulting equation is a contradiction.
Check if the solution satisfies all of the original equations.
Use a graphing calculator or computer software to check the solution.

Conclusion

Additional Resources

Final Thoughts

The linear combination method is a powerful tool for solving systems of linear equations. By following the steps outlined in this article, you can use the linear combination method to solve a wide range of problems in mathematics and other fields. Remember to be careful when applying the linear combination method, and avoid common mistakes such as multiplying the equations by the wrong constants and adding the constant terms incorrectly. With practice and patience, you can become proficient in using the linear combination method to solve systems of linear equations.