-y -10 = 6x 5x+y = -10 Elimination

Mar 11, 2025 by ADMIN 35 views

**Solving a System of Linear Equations using Elimination Method**

Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations is a fundamental concept in algebra and is used to find the values of the variables that satisfy all the equations in the system. In this article, we will discuss the elimination method, which is one of the most commonly used methods to solve a system of linear equations. We will use the given system of linear equations:

-2x - 10 = 6y 5x + y = -10

What is the Elimination Method?

The elimination method is a method used to solve a system of linear equations by adding or subtracting the equations to eliminate one of the variables. This method is based on the concept of combining like terms and using the properties of addition and subtraction to eliminate one of the variables.

Step 1: Write Down the Given Equations

The given system of linear equations is:

-2x - 10 = 6y 5x + y = -10

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. Let's multiply the first equation by 1 and the second equation by 6.

-2x - 10 = 6y 30x + 6y = -60

Step 3: Add or Subtract the Equations

Now, we can add or subtract the equations to eliminate one of the variables. Let's add the two equations.

(-2x - 10) + (30x + 6y) = 6y + (-60) 28x + 6y = -70

Step 4: Solve for One Variable

Now, we have a new equation with one variable. Let's solve for x.

28x = -70 - 6y x = (-70 - 6y) / 28

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

Now, we can substitute the value of x into one of the original equations to solve for the other variable. Let's substitute the value of x into the second original equation.

5x + y = -10 5((-70 - 6y) / 28) + y = -10

Step 6: Solve for the Other Variable

Now, we have an equation with one variable. Let's solve for y.

5((-70 - 6y) / 28) + y = -10 (-350 - 30y) / 28 + y = -10 (-350 - 30y + 28y) / 28 = -10 (-350 + 2y) / 28 = -10 -350 + 2y = -280 2y = 70 y = 35

Step 7: Find the Value of the Other Variable

Now, we have the value of y. Let's find the value of x.

x = (-70 - 6y) / 28 x = (-70 - 6(35)) / 28 x = (-70 - 210) / 28 x = -280 / 28 x = -10

Conclusion

In this article, we discussed the elimination method, which is one of the most commonly used methods to solve a system of linear equations. We used the given system of linear equations:

-2x - 10 = 6y 5x + y = -10

to demonstrate the steps involved in solving a system of linear equations using the elimination method. We found the values of x and y to be -10 and 35, respectively.

Example Problems

Here are some example problems to practice the elimination method:

Solve the system of linear equations:

2x + 3y = 7 x - 2y = -3

Solve the system of linear equations:

4x - 2y = 10 2x + 5y = 11

Discussion

The elimination method is a powerful tool for solving systems of linear equations. It is based on the concept of combining like terms and using the properties of addition and subtraction to eliminate one of the variables. The elimination method is particularly useful when the coefficients of the variables are integers.

However, the elimination method may not always be the most efficient method for solving systems of linear equations. In some cases, the substitution method may be more efficient. The substitution method involves substituting the value of one variable into one of the original equations to solve for the other variable.

Advantages and Disadvantages

The elimination method has several advantages, including:

It is a simple and straightforward method for solving systems of linear equations.
It is based on the concept of combining like terms, which makes it easy to understand and apply.
It is particularly useful when the coefficients of the variables are integers.

However, the elimination method also has some disadvantages, including:

It may not always be the most efficient method for solving systems of linear equations.
It requires the coefficients of the variables to be integers.
It may not be suitable for solving systems of linear equations with non-integer coefficients.

Real-World Applications

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

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

Q: What is the elimination method?

A: The elimination method is a method 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 determine which variable to eliminate, you need to look at the coefficients of the variables in the two equations. If the coefficients of one variable are the same, you can eliminate that variable by adding or subtracting the equations.

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

A: If the coefficients of the variables are not the same, you need to multiply one or both of the equations by a necessary multiple such that the coefficients of the variable to be eliminated are the same.

Q: How do I multiply the equations by a necessary multiple?

A: To multiply the equations by a necessary multiple, you need to multiply each term in the equation by the same number. For example, if you want to multiply the first equation by 2, you would multiply each term in the equation by 2.

Q: What if I get a negative value for one of the variables?

A: If you get a negative value for one of the variables, it means that the variable is not a solution to the system of linear equations. In this case, 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 a system of linear equations with three or more variables?

A: Yes, you can use the elimination method to solve a system of linear 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 multiplying the equations by a necessary multiple
Not adding or subtracting the equations correctly
Not checking your work for errors
Not using the correct method for solving the system of linear equations

Q: How do I know if the elimination method is the best method for solving a system of linear equations?

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

The complexity of the system of linear equations
The number of variables in the system of linear equations
The coefficients of the variables in the system of linear equations
The method of solution that you are most comfortable with

Q: Can I use the elimination method to solve a system of linear equations with non-integer coefficients?

A: No, the elimination method is not suitable for solving systems of linear equations with non-integer coefficients. In this case, you need to use a different method, such as the substitution method or the graphing method.

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

A: To check your work when using the elimination method, you need to:

Go back and check your work for errors
Verify that the solution satisfies both equations in the system of linear equations
Check that the solution is consistent with the given information

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

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

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

Conclusion

In conclusion, the elimination method is a powerful tool for solving systems of linear equations. It is based on the concept of combining like terms and using the properties of addition and subtraction to eliminate one of the variables. By following the steps outlined in this article, you can use the elimination method to solve systems of linear equations with ease. Remember to check your work carefully and verify that the solution satisfies both equations in the system of linear equations.