Solve The System Using Elimination.${ \begin{array}{r} -9x - 4y = 1 \ 3x + 3y = 3 \ ([?], \square) \end{array} }$Enter Your Answer In The Format ([?], \square).

Mar 9, 2025 by ADMIN 162 views

**Solving Systems of Linear Equations using Elimination Method**

Introduction

In mathematics, solving systems of linear equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. There are several methods to solve systems of linear equations, including substitution, elimination, and graphing. In this article, we will focus on the elimination method, which involves adding or subtracting equations to eliminate variables and solve for the remaining variables.

What is the Elimination Method?

The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate variables. This method involves creating a new equation by adding or subtracting the two original equations, which results in a single equation with one variable. The value of the variable can then be determined by solving the new equation.

Step-by-Step Guide to Solving Systems using Elimination

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

Write down the two equations: Write down the two linear equations that make up the system.
Identify the coefficients: Identify the coefficients of the variables in both equations.
Multiply the equations: Multiply both equations by necessary multiples such that the coefficients of the variables to be eliminated are the same.
Add or subtract the equations: Add or subtract the two equations to eliminate the variables.
Solve for the remaining variable: Solve for the remaining variable by dividing both sides of the equation by the coefficient of the variable.
Back-substitute: Back-substitute the value of the variable into one of the original equations to solve for the other variable.

Example: Solving the System using Elimination

Consider the following system of linear equations:

{ \begin{array}{r} -9x - 4y = 1 \\ 3x + 3y = 3 \\ ([?], \square) \end{array} \}

To solve this system using the elimination method, follow these steps:

Step 1: Write down the two equations

The two equations are:

-9x - 4y = 1 3x + 3y = 3

Step 2: Identify the coefficients

The coefficients of the variables are:

x: -9, 3 y: -4, 3

Step 3: Multiply the equations

To eliminate the variable x, multiply the first equation by 3 and the second equation by 9:

3(-9x - 4y) = 3(1) 9(3x + 3y) = 9(3)

This results in:

-27x - 12y = 3 27x + 27y = 27

Step 4: Add or subtract the equations

Add the two equations to eliminate the variable x:

(-27x - 12y) + (27x + 27y) = 3 + 27 -12y + 27y = 30 15y = 30

Step 5: Solve for the remaining variable

Divide both sides of the equation by 15 to solve for y:

y = 30/15 y = 2

Step 6: Back-substitute

Back-substitute the value of y into one of the original equations to solve for x. Using the first equation:

-9x - 4y = 1 -9x - 4(2) = 1 -9x - 8 = 1 -9x = 9 x = -1

Therefore, the solution to the system is:

([?], \square) = (-1, 2)

Conclusion

Solving systems of linear equations using the elimination method involves adding or subtracting equations to eliminate variables and solve for the remaining variables. By following the step-by-step guide outlined in this article, you can solve systems of linear equations using the elimination method. Remember to identify the coefficients, multiply the equations, add or subtract the equations, solve for the remaining variable, and back-substitute to find the solution to the system.

Common Mistakes to Avoid

When solving systems of linear equations using the elimination method, there are several common mistakes to avoid:

Incorrectly identifying the coefficients: Make sure to identify the coefficients of the variables correctly.
Incorrectly multiplying the equations: Make sure to multiply the equations by the correct multiples.
Incorrectly adding or subtracting the equations: Make sure to add or subtract the equations correctly.
Incorrectly solving for the remaining variable: Make sure to solve for the remaining variable correctly.
Incorrectly back-substituting: Make sure to back-substitute the value of the variable correctly.

Real-World Applications

Solving systems of linear equations using the elimination method has several real-world applications, including:

Physics: Solving systems of linear equations is used to model real-world problems in physics, such as motion and forces.
Engineering: Solving systems of linear equations is used to design and optimize systems in engineering, such as electrical and mechanical systems.
Economics: Solving systems of linear equations is used to model real-world problems in economics, such as supply and demand.

Conclusion

Introduction

In our previous article, we discussed the elimination method for solving systems of linear equations. In this article, we will answer some frequently asked questions about the elimination method and provide additional examples to help you understand the concept better.

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 variables.

Q: How do I choose which variable to eliminate?

A: To choose which variable to eliminate, look for the coefficients of the variables in both equations. If the coefficients are the same, you can eliminate one of the variables by adding or subtracting the equations.

Q: What if the coefficients are not the same?

A: If the coefficients are not the same, you can multiply one or both of the equations by a multiple of the coefficient to make the coefficients the same.

Q: How do I add or subtract the equations?

A: To add or subtract the equations, add or subtract the corresponding terms of the two equations.

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. 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 systems with more than two equations?

A: Yes, you can use the elimination method to solve systems with more than two equations. However, you will need to use a combination of the elimination method and other techniques, such as substitution or graphing, to solve the system.

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:

Incorrectly identifying the coefficients: Make sure to identify the coefficients of the variables correctly.
Incorrectly multiplying the equations: Make sure to multiply the equations by the correct multiples.
Incorrectly adding or subtracting the equations: Make sure to add or subtract the equations correctly.
Incorrectly solving for the remaining variable: Make sure to solve for the remaining variable correctly.
Incorrectly back-substituting: Make sure to back-substitute the value of the variable correctly.

Q: Can I use the elimination method to solve systems with fractions or decimals?

A: Yes, you can use the elimination method to solve systems with fractions or decimals. However, you will need to follow the same steps as before, but with the added complexity of fractions or decimals.

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

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

Physics: Solving systems of linear equations is used to model real-world problems in physics, such as motion and forces.
Engineering: Solving systems of linear equations is used to design and optimize systems in engineering, such as electrical and mechanical systems.
Economics: Solving systems of linear equations is used to model real-world problems in economics, such as supply and demand.

Conclusion

The elimination method is a powerful tool for solving systems of linear equations. By following the step-by-step guide outlined in this article, you can solve systems of linear equations using the elimination method. Remember to identify the coefficients, multiply the equations, add or subtract the equations, solve for the remaining variable, and back-substitute to find the solution to the system.

Additional Examples

Here are some additional examples of solving systems of linear equations using the elimination method:

Example 1:

{ \begin{array}{r} 2x + 3y = 7 \\ x - 2y = -3 \\ ([?], \square) \end{array} \}

To solve this system using the elimination method, follow these steps:

Multiply the first equation by 2 and the second equation by 3: 2(2x + 3y) = 2(7) 3(x - 2y) = 3(-3) 4x + 6y = 14 3x - 6y = -9
Add the two equations to eliminate the variable y: (4x + 6y) + (3x - 6y) = 14 + (-9) 7x = 5
Solve for x: x = 5/7
Back-substitute the value of x into one of the original equations to solve for y: 2x + 3y = 7 2(5/7) + 3y = 7 10/7 + 3y = 7 3y = 49/7 - 10/7 3y = 39/7 y = 13/7

Therefore, the solution to the system is:

([?], \square) = (5/7, 13/7)

Example 2:

{ \begin{array}{r} x + 2y = 4 \\ 3x - 2y = 5 \\ ([?], \square) \end{array} \}

To solve this system using the elimination method, follow these steps:

Multiply the first equation by 3 and the second equation by 1: 3(x + 2y) = 3(4) (3x - 2y) = 5 3x + 6y = 12 3x - 2y = 5
Subtract the second equation from the first equation to eliminate the variable x: (3x + 6y) - (3x - 2y) = 12 - 5 8y = 7
Solve for y: y = 7/8
Back-substitute the value of y into one of the original equations to solve for x: x + 2y = 4 x + 2(7/8) = 4 x + 7/4 = 4 x = 4 - 7/4 x = 3/4

Therefore, the solution to the system is:

([?], \square) = (3/4, 7/8)