Solve The System Of Equations By Elimination:$\[ \begin{cases} -x + 6y = -1 \\ -3x - 6y = 15 \end{cases} \\]Solution: $\square$ . $\square$

Feb 28, 2025 by ADMIN 143 views

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Introduction

In this article, we will explore the method of elimination to solve a system of linear equations. The elimination method is a powerful technique used to solve systems of 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 can be easily simplified to integers.

What is the Elimination Method?

The elimination method involves adding or subtracting equations to eliminate one of the variables. This is done by multiplying the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same, but with opposite signs. Once the coefficients are the same, we can add or subtract the equations to eliminate the variable.

Step 1: Write Down the Given Equations

The given system of equations is:

{ \begin{cases} -x + 6y = -1 \\ -3x - 6y = 15 \end{cases} \}

Step 2: Multiply the Equations by Necessary Multiples

To eliminate the variable x, we need to multiply the first equation by 3 and the second equation by 1. This will give us:

{ \begin{cases} -3x + 18y = -3 \\ -3x - 6y = 15 \end{cases} \}

Step 3: Add or Subtract the Equations

Now that the coefficients of x are the same, we can add the two equations to eliminate x:

{ -3x + 18y = -3 \\ -3x - 6y = 15 \\ ------------------- 0 + 12y = 12 \}

Step 4: Solve for y

Now that we have eliminated x, we can solve for y by dividing both sides of the equation by 12:

{ 12y = 12 \\ y = 12/12 \\ y = 1 \}

Step 5: Substitute y into One of the Original Equations

Now that we have found y, we can substitute it into one of the original equations to find x. Let's use the first equation:

{ -x + 6y = -1 \\ -x + 6(1) = -1 \\ -x + 6 = -1 \\ -x = -7 \\ x = 7 \}

Step 6: Write Down the Solution

The solution to the system of equations is x = 7 and y = 1.

Conclusion

In this article, we have used the elimination method to solve a system of linear equations. We have shown that by multiplying the equations by necessary multiples and adding or subtracting them, we can eliminate one of the variables and solve for the other variable. This method is particularly useful when the coefficients of the variables in the equations are integers or can be easily simplified to integers.

Discussion

The elimination method is a powerful technique used to solve systems of equations. It is particularly useful when the coefficients of the variables in the equations are integers or can be easily simplified to integers. However, it may not be as effective when the coefficients are fractions or decimals.

Example Problems

Here are a few example problems that can be solved using the elimination method:

\begin{cases} 2x + 3y = 7 \ 4x - 2y = -3 \end{cases} }$

\begin{cases} x + 2y = 4 \ 3x - 2y = 5 \end{cases} }$

\begin{cases} 2x - 3y = 5 \ x + 2y = 3 \end{cases} }$

Tips and Tricks

Here are a few tips and tricks 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, but with opposite signs.
Be careful when adding or subtracting the equations to eliminate the variable.
Make sure to solve for the variable that is eliminated first.
Use the elimination method when the coefficients of the variables in the equations are integers or can be easily simplified to integers.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "College Algebra" by James Stewart
[3] "Linear Algebra and Its Applications" by Gilbert Strang

Q: What is the elimination method?

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

Q: When should I use the elimination method?

A: You should use the elimination method when the coefficients of the variables in the equations are integers or can be easily simplified to integers.

Q: How do I multiply the equations by necessary multiples?

A: To multiply the equations by necessary multiples, you need to multiply the coefficients of the variable to be eliminated by the same number, but with opposite signs.

Q: What if the coefficients of the variables are fractions or decimals?

A: If the coefficients of the variables are fractions or decimals, you may need to multiply the equations by a common multiple to eliminate the variable.

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

A: To add or subtract the equations, you need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same, but with opposite signs.

Q: What if I get a zero on one side of the equation?

A: If you get a zero on one side of the equation, you can ignore it and solve for the other variable.

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, but it may be more complicated.

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 multiplying the equations by necessary multiples
Not adding or subtracting the equations correctly
Not solving for the variable that is eliminated first
Not checking the solution for consistency

Q: How do I check the solution for consistency?

A: To check the solution for consistency, you need to plug the values of the variables back into the original equations and check if they are true.

Q: What if the solution is not consistent?

A: If the solution is not consistent, you need to go back and check your work to see where you made a mistake.

Q: Can I use the elimination method to solve systems of equations with non-linear equations?

A: No, the elimination method is only used to solve systems of linear equations.

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 and engineering
Solving systems of equations in economics and finance
Solving systems of equations in computer science and data analysis

Q: How do I practice using the elimination method?

A: You can practice using the elimination method by working through example problems and exercises in a textbook or online resource.

Q: What are some online resources for learning about the elimination method?

A: Some online resources for learning about the elimination method include:

Khan Academy: Systems of Equations
MIT OpenCourseWare: Linear Algebra
Wolfram MathWorld: Systems of Linear Equations

Q: Can I use the elimination method to solve systems of equations with complex numbers?

A: Yes, you can use the elimination method to solve systems of equations with complex numbers.

Q: What are some tips for using the elimination method with complex numbers?

A: Some tips for using the elimination method with complex numbers include:

Make sure to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same, but with opposite signs.
Be careful when adding or subtracting the equations to eliminate the variable.
Make sure to solve for the variable that is eliminated first.
Use the elimination method when the coefficients of the variables in the equations are complex numbers or can be easily simplified to complex numbers.