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Introduction

The addition method is a technique used to solve systems of linear equations. It involves adding two or more equations together to eliminate one of the variables. In this article, we will use the addition method to solve a system of two linear equations.

The System of Equations

The system of equations we will be solving is:

{ \begin{array}{l} 4x + 4y = 20 \\ 2x - 4y = 4 \end{array} \}

Step 1: Multiply the Equations

To use the addition method, we need to multiply the equations by necessary multiples such that the coefficients of y's in both equations are the same. We can multiply the first equation by 1 and the second equation by 1.

Equation 1:

4x + 4y = 20

Equation 2:

2x - 4y = 4

Step 2: Add the Equations

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

(4x + 4y) + (2x - 4y) = 20 + 4

Simplifying the equation, we get:

6x = 24

Step 3: Solve for x

Now, we solve for x by dividing both sides of the equation by 6.

x = 24/6
x = 4

Step 4: Substitute x into One of the Original Equations

Now that we have the value of x, we can substitute it into one of the original equations to solve for y. We will use the first equation.

4x + 4y = 20
4(4) + 4y = 20
16 + 4y = 20
4y = 20 - 16
4y = 4
y = 4/4
y = 1

The Solution

The solution to the system of equations is x = 4 and y = 1.

Conclusion

In this article, we used the addition method to solve a system of two linear equations. We multiplied the equations by necessary multiples, added the equations together, solved for x, and then substituted x into one of the original equations to solve for y. The solution to the system of equations is x = 4 and y = 1.

Example Problems

Here are a few example problems that you can try using the addition method:

  1. Solve the system of equations:

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

  1. Solve the system of equations:

{ \begin{array}{l} x + 2y = 6 \\ 3x - 2y = 10 \end{array} \}

  1. Solve the system of equations:

{ \begin{array}{l} 4x - 3y = 11 \\ 2x + 5y = 17 \end{array} \}

Tips and Tricks

Here are a few tips and tricks that you can use when solving systems of equations using the addition method:

  1. Make sure to multiply the equations by necessary multiples before adding them together.
  2. Simplify the equation after adding the two equations together.
  3. Solve for one variable before substituting it into one of the original equations.
  4. Check your solution by plugging it back into the original equations.

Common Mistakes

Here are a few common mistakes that you can make when solving systems of equations using the addition method:

  1. Not multiplying the equations by necessary multiples before adding them together.
  2. Not simplifying the equation after adding the two equations together.
  3. Not solving for one variable before substituting it into one of the original equations.
  4. Not checking your solution by plugging it back into the original equations.

Conclusion

Introduction

In our previous article, we used the addition method to solve a system of two linear equations. In this article, we will answer some frequently asked questions about the addition method and provide additional examples to help you understand the concept better.

Q: What is the addition method?

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

Q: How do I know which equations to add together?

A: To use the addition method, you need to multiply the equations by necessary multiples such that the coefficients of the variable you want to eliminate are the same. Then, you add the two equations together.

Q: What if the coefficients of the variable I want to eliminate are not the same?

A: If the coefficients of the variable you want to eliminate are not the same, you need to multiply the equations by necessary multiples to make the coefficients the same. For example, if you have the equations:

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

You can multiply the first equation by 1 and the second equation by 3 to make the coefficients of y the same.

Q: How do I simplify the equation after adding the two equations together?

A: After adding the two equations together, you need to simplify the equation by combining like terms. For example, if you have the equation:

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

You can simplify the equation by combining like terms:

5x+y=4{ 5x + y = 4 }

Q: How do I solve for one variable?

A: To solve for one variable, you need to isolate the variable on one side of the equation. For example, if you have the equation:

5x+y=4{ 5x + y = 4 }

You can solve for x by subtracting y from both sides of the equation:

5x=4−y{ 5x = 4 - y }

Then, you can divide both sides of the equation by 5 to solve for x:

x=4−y5{ x = \frac{4 - y}{5} }

Q: How do I check my solution?

A: To check your solution, you need to plug it back into the original equations. For example, if you have the solution x = 4 and y = 1, you can plug it back into the original equations:

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

You can substitute x = 4 and y = 1 into the first equation:

2(4)+3(1)=7{ 2(4) + 3(1) = 7 }

Simplifying the equation, you get:

8+3=7{ 8 + 3 = 7 }

Which is true. You can also substitute x = 4 and y = 1 into the second equation:

4−2(1)=−3{ 4 - 2(1) = -3 }

Simplifying the equation, you get:

4−2=−3{ 4 - 2 = -3 }

Which is also true.

Example Problems

Here are a few example problems that you can try using the addition method:

  1. Solve the system of equations:

{ \begin{array}{l} x + 2y = 6 \\ 3x - 2y = 10 \end{array} \}

  1. Solve the system of equations:

{ \begin{array}{l} 4x - 3y = 11 \\ 2x + 5y = 17 \end{array} \}

  1. Solve the system of equations:

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

Tips and Tricks

Here are a few tips and tricks that you can use when solving systems of equations using the addition method:

  1. Make sure to multiply the equations by necessary multiples before adding them together.
  2. Simplify the equation after adding the two equations together.
  3. Solve for one variable before substituting it into one of the original equations.
  4. Check your solution by plugging it back into the original equations.

Common Mistakes

Here are a few common mistakes that you can make when solving systems of equations using the addition method:

  1. Not multiplying the equations by necessary multiples before adding them together.
  2. Not simplifying the equation after adding the two equations together.
  3. Not solving for one variable before substituting it into one of the original equations.
  4. Not checking your solution by plugging it back into the original equations.

Conclusion

In conclusion, the addition method is a powerful technique for solving systems of linear equations. By multiplying the equations by necessary multiples, adding the equations together, solving for one variable, and then substituting it into one of the original equations, we can solve systems of equations with ease. Remember to simplify the equation after adding the two equations together, solve for one variable before substituting it into one of the original equations, and check your solution by plugging it back into the original equations.