Example of the Addition Method

Oct 13, 2025 by leolavanderia.com.br 2 views

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Introduction

The addition method is a technique used to solve systems of linear equations. It involves adding two equations together to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the two equations are additive inverses of each other. In this article, we will discuss how to solve a system of equations using the addition method.

What is the Addition Method?

The addition method is a technique used to solve systems of linear equations. It involves adding two equations together to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the two equations are additive inverses of each other. The addition method is also known as the elimination method.

How Does the Addition Method Work?

The addition method works by adding two equations together to eliminate one of the variables. This is done by multiplying one or both of the equations by a constant, and then adding the two equations together. The resulting equation will have only one variable, which can be solved for.

Example of the Addition Method

Let's consider the following system of equations:

{ \begin{cases} 4x + y = 19 \\ 3x - y = 9 \end{cases} \}

To solve this system using the addition method, we need to add the two equations together. However, the coefficients of the variables in the two equations are not additive inverses of each other. Therefore, we need to multiply one of the equations by a constant to make the coefficients additive inverses.

Multiplying the Equations by Constants

Let's multiply the first equation by 1 and the second equation by 1. This will give us:

{ \begin{cases} 4x + y = 19 \\ 3x - y = 9 \end{cases} \}

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

Adding the Equations Together

Let's add the two equations together:

{ (4x + y) + (3x - y) = 19 + 9 \}

Simplifying the equation, we get:

{ 7x = 28 \}

Now, we can solve for x.

Solving for x

To solve for x, we need to divide both sides of the equation by 7:

{ x = \frac{28}{7} \}

Simplifying the equation, we get:

{ x = 4 \}

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y.

Substituting x into the Original Equation

Let's substitute x into the first original equation:

{ 4x + y = 19 \}

Substituting x = 4, we get:

{ 4(4) + y = 19 \}

Simplifying the equation, we get:

{ 16 + y = 19 \}

Now, we can solve for y.

Solving for y

To solve for y, we need to subtract 16 from both sides of the equation:

{ y = 19 - 16 \}

Simplifying the equation, we get:

{ y = 3 \}

Therefore, the solution to the system of equations is x = 4 and y = 3.

Conclusion

The addition method is a technique used to solve systems of linear equations. It involves adding two equations together to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the two equations are additive inverses of each other. In this article, we discussed how to solve a system of equations using the addition method.

Example Problems

Problem 1

Solve the system of equations using the addition method:

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

Solution

Let's multiply the first equation by 1 and the second equation by 1. This will give us:

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

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

Let's add the two equations together:

{ (2x + 3y) + (x - 2y) = 7 + (-3) \}

Simplifying the equation, we get:

{ 3x + y = 4 \}

Now, we can solve for x.

Let's solve for x:

{ 3x + y = 4 \}

Subtracting y from both sides of the equation, we get:

{ 3x = 4 - y \}

Dividing both sides of the equation by 3, we get:

{ x = \frac{4 - y}{3} \}

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y.

Let's substitute x into the first original equation:

{ 2x + 3y = 7 \}

Substituting x = (4 - y)/3, we get:

{ 2(\frac{4 - y}{3}) + 3y = 7 \}

Simplifying the equation, we get:

{ \frac{8 - 2y}{3} + 3y = 7 \}

Multiplying both sides of the equation by 3, we get:

{ 8 - 2y + 9y = 21 \}

Simplifying the equation, we get:

{ 7y = 13 \}

Dividing both sides of the equation by 7, we get:

{ y = \frac{13}{7} \}

Therefore, the solution to the system of equations is x = (4 - y)/3 and y = 13/7.

Problem 2

Solve the system of equations using the addition method:

{ \begin{cases} x + 2y = 6 \\ 2x - 3y = -2 \end{cases} \}

Solution

Let's multiply the first equation by 1 and the second equation by 1. This will give us:

{ \begin{cases} x + 2y = 6 \\ 2x - 3y = -2 \end{cases} \}

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

Let's add the two equations together:

{ (x + 2y) + (2x - 3y) = 6 + (-2) \}

Simplifying the equation, we get:

{ 3x - y = 4 \}

Now, we can solve for x.

Let's solve for x:

{ 3x - y = 4 \}

Adding y to both sides of the equation, we get:

{ 3x = 4 + y \}

Dividing both sides of the equation by 3, we get:

{ x = \frac{4 + y}{3} \}

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y.

Let's substitute x into the first original equation:

{ x + 2y = 6 \}

Substituting x = (4 + y)/3, we get:

{ \frac{4 + y}{3} + 2y = 6 \}

Simplifying the equation, we get:

{ \frac{4 + y}{3} + \frac{6y}{3} = 6 \}

Multiplying both sides of the equation by 3, we get:

{ 4 + y + 6y = 18 \}

Simplifying the equation, we get:

{ 7y = 14 \}

Dividing both sides of the equation by 7, we get:

{ y = \frac{14}{7} \}

Therefore, the solution to the system of equations is x = (4 + y)/3 and y = 2.

Conclusion

The addition method is a technique used to solve systems of linear equations. It involves adding two equations together to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the two equations are additive inverses of each other. In this article, we discussed how to solve a system of equations using the addition method.

References

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

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Introduction

In our previous article, we discussed how to solve systems of equations using the addition method. The addition method is a technique used to solve systems of linear equations by adding two equations together to eliminate one of the variables. In this article, we will answer some frequently asked questions about solving systems of equations by the addition method.

Q&A

Q: What is the addition method?

A: The addition method is a technique used to solve systems of linear equations by adding two equations together to eliminate one of the variables.

Q: When can I use the addition method?

A: You can use the addition method when the coefficients of the variables in the two equations are additive inverses of each other.

Q: How do I know if the coefficients are additive inverses?

A: To determine if the coefficients are additive inverses, you can add the coefficients of the variables in the two equations together. If the sum is zero, then the coefficients are additive inverses.

Q: What if the coefficients are not additive inverses?

A: If the coefficients are not additive inverses, you can multiply one of the equations by a constant to make the coefficients additive inverses.

Q: How do I multiply an equation by a constant?

A: To multiply an equation by a constant, you can multiply each term in the equation by the constant.

Q: What if I multiply an equation by a constant and the coefficients are still not additive inverses?

A: If the coefficients are still not additive inverses after multiplying an equation by a constant, you can try multiplying the equation by a different constant.

Q: How do I add two equations together?

A: To add two equations together, you can add the corresponding terms in the two equations.

Q: What if I add two equations together and the resulting equation has a variable with a coefficient of zero?

A: If the resulting equation has a variable with a coefficient of zero, you can solve for the other variable.

Q: How do I solve for a variable?

A: To solve for a variable, you can isolate the variable on one side of the equation.

Q: What if I have two variables and I want to solve for both of them?

A: If you have two variables and you want to solve for both of them, you can use the addition method to eliminate one of the variables, and then solve for the other variable.

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

A: No, the addition method can only be used to solve systems of equations with two variables.

Q: What if I have a system of equations with more than two variables and I want to solve for all of the variables?

A: If you have a system of equations with more than two variables and you want to solve for all of the variables, you can use a different method, such as substitution or elimination.

Conclusion

The addition method is a technique used to solve systems of linear equations by adding two equations together to eliminate one of the variables. In this article, we answered some frequently asked questions about solving systems of equations by the addition method.

Example Problems

Problem 1

Solve the system of equations using the addition method:

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

Solution

Let's multiply the first equation by 1 and the second equation by 1. This will give us:

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

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

Let's add the two equations together:

{ (2x + 3y) + (x - 2y) = 7 + (-3) \}

Simplifying the equation, we get:

{ 3x + y = 4 \}

Now, we can solve for x.

Let's solve for x:

{ 3x + y = 4 \}

Adding y to both sides of the equation, we get:

{ 3x = 4 + y \}

Dividing both sides of the equation by 3, we get:

{ x = \frac{4 + y}{3} \}

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y.

Let's substitute x into the first original equation:

{ 2x + 3y = 7 \}

Substituting x = (4 + y)/3, we get:

{ 2(\frac{4 + y}{3}) + 3y = 7 \}

Simplifying the equation, we get:

{ \frac{8 + 2y}{3} + 3y = 7 \}

Multiplying both sides of the equation by 3, we get:

{ 8 + 2y + 9y = 21 \}

Simplifying the equation, we get:

{ 11y = 13 \}

Dividing both sides of the equation by 11, we get:

{ y = \frac{13}{11} \}

Therefore, the solution to the system of equations is x = (4 + y)/3 and y = 13/11.

Problem 2

Solve the system of equations using the addition method:

{ \begin{cases} x + 2y = 6 \\ 2x - 3y = -2 \end{cases} \}

Solution

Let's multiply the first equation by 1 and the second equation by 1. This will give us:

{ \begin{cases} x + 2y = 6 \\ 2x - 3y = -2 \end{cases} \}

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

Let's add the two equations together:

{ (x + 2y) + (2x - 3y) = 6 + (-2) \}

Simplifying the equation, we get:

{ 3x - y = 4 \}

Now, we can solve for x.

Let's solve for x:

{ 3x - y = 4 \}

Adding y to both sides of the equation, we get:

{ 3x = 4 + y \}

Dividing both sides of the equation by 3, we get:

{ x = \frac{4 + y}{3} \}

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y.

Let's substitute x into the first original equation:

{ x + 2y = 6 \}

Substituting x = (4 + y)/3, we get:

{ \frac{4 + y}{3} + 2y = 6 \}

Simplifying the equation, we get:

{ \frac{4 + y}{3} + \frac{6y}{3} = 6 \}

Multiplying both sides of the equation by 3, we get:

{ 4 + y + 6y = 18 \}

Simplifying the equation, we get:

{ 7y = 14 \}

Dividing both sides of the equation by 7, we get:

{ y = \frac{14}{7} \}

Therefore, the solution to the system of equations is x = (4 + y)/3 and y = 2.