Cross multiplication method of solving a pair of linear equations

Cross Multiplication Method for Solving a Pair of Linear Equations

The cross multiplication method is a powerful algebraic technique used to solve a pair of linear equations in two variables, especially when the equations are in standard form. It eliminates the need for step-by-step substitution or elimination by directly applying a formula involving determinants.

Standard Form of Linear Equations

The method is applicable when both equations are written as:

a₁x + b₁y + c₁ = 0  
a₂x + b₂y + c₂ = 0
  

Here, a₁, b₁, and c₁ are the coefficients from the first equation, and a₂, b₂, and c₂ are from the second.

Cross Multiplication Formula

To solve the equations using cross multiplication, use the following setup:

x      =   (b₁·c₂ - b₂·c₁) / (a₁·b₂ - a₂·b₁)  
y      =   (c₁·a₂ - c₂·a₁) / (a₁·b₂ - a₂·b₁)
  

The denominator in both expressions is the same and must not be zero (as division by zero is undefined).

Steps to Solve Using Cross Multiplication

1. Write both equations in standard form with all terms on the left-hand side.
2. Identify the coefficients a₁, b₁, c₁ and a₂, b₂, c₂.
3. Substitute them into the cross multiplication formula.
4. Calculate the values of x and y.

Example:

Solve the system using cross multiplication:

2x + 3y = 17  
4x − 5y = −1
  

Step 1: Rewrite in standard form:

2x + 3y − 17 = 0  
4x − 5y + 1 = 0
  

Step 2: Identify coefficients:

• a₁ = 2, b₁ = 3, c₁ = −17
• a₂ = 4, b₂ = −5, c₂ = 1

Step 3: Apply the formula:

x = (3×1 − (−5)×(−17)) / (2×(−5) − 4×3)  
  = (3 − 85) / (−10 − 12)  
  = (−82) / (−22)  
  = 41 / 11

y = ((−17)×4 − 1×2) / (2×(−5) − 4×3)  
  = (−68 − 2) / (−22)  
  = (−70) / (−22)  
  = 35 / 11
  

Final Answer: x = 41⁄11, y = 35⁄11

Advantages of the Cross Multiplication Method

• Fast and systematic when equations are in standard form.
• Does not require solving for one variable first.
• Useful when coefficients are easily manageable.

Limitations

• Equations must be arranged in standard form.
• Less convenient for equations with fractions or decimals.
• Does not work if the determinant (denominator) equals zero (i.e., the equations are either parallel or overlapping).

Conclusion

The cross multiplication method is an effective algebraic approach to solving a pair of linear equations in two variables. It avoids step-by-step manipulation and offers a direct route to the solution. Nonetheless, this method is most effective when the equations are arranged clearly in standard form and the coefficients are simple to work with.