Elimination method of solving a pair of linear equations

Elimination Method for Solving a Pair of Linear Equations

The elimination method is a widely used algebraic technique to solve a pair of linear equations in two variables. It works by adjusting and combining the equations to remove one variable, which simplifies the process of finding the value of the remaining one. This technique is particularly effective when the equations are already expressed in standard linear format.

Standard Form of Linear Equations

The elimination method applies to equations written as:

a₁x + b₁y = c₁  
a₂x + b₂y = c₂  
  

In these equations, a₁, b₁, c₁ and a₂, b₂, c₂ represent constant values.

Steps in the Elimination Method

1. Arrange both equations in standard form.
2. Multiply one or both equations (if needed) so that the coefficients of one variable become equal or opposites.
3. Add or subtract the equations to eliminate one variable.
4. Find the value of the second variable by solving the simplified equation.
5. Plug this value into one of the initial equations to determine the other variable.

Example:

Let's solve the following system of equations:

2x + 3y = 12   ...(1)  
4x − 3y = 6    ...(2)
  

Step 1: Notice that the coefficients of y in both equations are additive inverses (3y and −3y).

Step 2: Add both equations:

(2x + 3y) + (4x − 3y) = 12 + 6  
⇒ 6x = 18  
⇒ x = 3
  

Step 3: Substitute x = 3 into equation (1):

2(3) + 3y = 12  
6 + 3y = 12  
3y = 6  
y = 2
  

Solution: x = 3, y = 2

Advantages of the Elimination Method

• Efficient when coefficients of a variable are easily made equal or opposite.
• Eliminates one variable directly, reducing the complexity of solving the system.
• Suitable for problems with integers and manageable coefficients.

Limitations

• Requires multiplication of equations if coefficients don’t align initially.
• May become lengthy if the numbers involved are large or fractions.

The elimination method offers a systematic and dependable way to solve linear equations involving two variables. By strategically eliminating one variable, it simplifies the process of finding the solution to the system. With practice, this method becomes an essential tool in algebra and real-world problem solving.