How to solve linear equations by elimination?

The elimination method, also known as the addition method, solves systems of linear equations by adding or subtracting the equations to eliminate one variable. Here's a breakdown of the process:

Steps to Solve Linear Equations by Elimination

1. Arrange the Equations: Ensure that both equations are written in the standard form, typically Ax + By = C, where A, B, and C are constants, and x and y are variables.

2. Identify the Variable to Eliminate: Look for a variable that has either the same coefficient or coefficients that are easy to make the same (or opposite) by multiplication.

3. Multiply (if necessary): If needed, multiply one or both equations by a constant so that the coefficients of the variable you want to eliminate are either the same or opposites.

4. Add or Subtract the Equations:

   • If the coefficients are the same, subtract one equation from the other.
   • If the coefficients are opposites, add the equations.

   This step will eliminate one variable, resulting in a new equation with only one variable.

5. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.

6. Substitute: Substitute the value found in step 5 back into either of the original equations to solve for the other variable.

7. Check Your Solution: Substitute both values (x and y) into both original equations to ensure they satisfy both equations.

Example:

Let's use the example provided in the reference:

• 2x + y = 5
• 3x - y = 5

Here, we can directly add the equations to eliminate y because the y coefficients are already opposites (+1 and -1).

1. Add the Equations:
   (2x + y) + (3x - y) = 5 + 5
   5x = 10

2. Solve for x:
   x = 10 / 5
   x = 2

3. Substitute x = 2 into the first equation:
   2(2) + y = 5
   4 + y = 5
   y = 5 - 4
   y = 1

4. Solution:
   x = 2, y = 1

Therefore, the solution to the system of equations is x = 2 and y = 1.