Comparison method of solving a pair of linear equations

Comparison Method for Solving a Pair of Linear Equations

The comparison method is an algebraic technique used to solve a pair of linear equations in two variables. It involves expressing both equations in terms of the same variable (usually y or x), then comparing the resulting expressions to find the value of the other variable.

When to Use the Comparison Method

This method is especially useful when both equations can easily be written in terms of the same variable. It allows for direct comparison and eliminates the need for substitution or elimination steps.

Standard Form of Linear Equations

The method is typically applied to equations written in the form:

a₁x + b₁y = c₁  
a₂x + b₂y = c₂
  

Steps to Use the Comparison Method

1. Rearrange both equations to express the same variable (x or y) in terms of the other.
2. Set the two expressions equal to each other since they represent the same variable.
3. Solve the resulting equation to find the value of one variable.
4. Substitute the value into one of the original equations to find the other variable.

Example:

Solve the system of equations using the comparison method:

x + 2y = 8        ...(1)  
3x − y = 5        ...(2)
  

Step 1: Rearrange both equations to isolate x on one side.

From (1): x = 8 − 2y  
From (2): x = (5 + y)/3
  

Step 2: Compare the two expressions for x:

8 − 2y = (5 + y)/3
  

Step 3: Clear the denominator by multiplying both sides of the equation by 3.

3(8 − 2y) = 5 + y  
24 − 6y = 5 + y  
24 − 5 = 6y + y  
19 = 7y ⇒ y = 19/7
  

Step 4: Substitute y = 19⁄7 into x = 8 − 2y:

x = 8 − 2(19/7) = 8 − 38/7  
x = (56 − 38)/7 = 18/7
  

Final solution: x = 18⁄7, y = 19⁄7

Advantages of the Comparison Method

• Simple to apply when both equations can be rearranged easily.
• Provides a direct way to equate and solve expressions.
• Reduces algebraic manipulation in some cases.

Limitations

• Can lead to complicated fractions or long calculations if the equations are difficult to rearrange.
• Less effective for equations with variables that don’t isolate easily.

Conclusion

The comparison method offers a straightforward approach for solving linear equations when expressions for a variable can be easily derived. Though not always the most efficient method, it is useful in cases where direct comparison leads to quick results.