Solving Systems of Linear Equations Using the Elimination Method: Addition Method

Solving Systems of Linear Equations Using the Elimination Method: Addition Method

The elimination method is a technique used to solve systems of linear equations where the solution involves eliminating variables or adding equations to obtain an equation in one variable. In this article, we will focus on the addition method of solving systems of linear equations using the elimination method.

Solving Systems of Linear Equations by Elimination

The elimination method is a systematic approach for solving systems of linear equations. The steps involved in solving a system of linear equations using the elimination method include:

1. Write both equations in standard form. If any coefficients are fractions, clear them.
2. Make the coefficients of one variable opposites.
3. Decide which variable you will eliminate.
4. Multiply one or both equations so that the coefficients of that variable are opposites.
5. Add the equations resulting from Step 2 to eliminate one variable.
6. Solve for the remaining variable.
7. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.
8. Write the solution as an ordered pair.

Addition Method

The addition method is a specific application of the elimination method where we add the equations to eliminate one of the variables. Here's how to solve a system of linear equations using the addition method:

1. Write both equations in standard form.
2. If necessary, multiply one or both equations so that the coefficients of one variable are opposites.
3. Add the equations to eliminate one variable.
4. Solve for the remaining variable.
5. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable.
6. Write the solution as an ordered pair.

Let's look at an example to understand this method better.

Example

Consider the following system of linear equations:

1. 3x + 4y = 12
2. 2x - y = 6

To solve this system using the addition method, we will follow these steps:

1. Write both equations in standard form. In this case, the equations are already in standard form.
2. Multiply the second equation by 4 to make the coefficients of y opposites: 8x - 4y = 24.
3. Add the two equations to eliminate the y variable: 3x + 4y + 8x - 4y = 12 + 24.
4. Simplify and solve for x: 11x = 36.
5. Solve for x: x = 36/11 ≈ 3.27.
6. Substitute x = 3.27 into one of the original equations to solve for y: 3(3.27) + 4y = 12.
7. Solve for y: 4y = 12 - 9.81.
8. Solve for y: y = 2.19.

The solution is x = 3.27 and y = 2.19.

Practice

Now that you understand the addition method, try solving the following system of linear equations using the elimination method:

1. 5x + 3y = 18
2. 2x - y = 6

Remember to follow the steps outlined in the elimination method: write both equations in standard form, make the coefficients of one variable opposites, decide which variable you will eliminate, multiply one or both equations so that the coefficients of that variable are opposites, add the equations resulting from Step 2 to eliminate one variable, solve for the remaining variable, substitute the solution from Step 4 into one of the original equations, and finally, write the solution as an ordered pair.