Unique Solution by Cross Multiplication Method

We get a unique solution to the pair of linear equations if the given pair of lines are consistent. For any two lines a₁x + b₁y = -c₁ and a₂x + b₂y = -c₂ to be consistent the required condition is,

a₁/a₂ ≠ b₁/b₂

This situation can also be represented as,

a₁b₁ ≠ a₂b₂

a₁b₁ – a₂b₂ = 0

In case of a unique solution, the line for two equations will intersect at a unique point which is the solution of the two equations.

For example, Check whether 2x + 3y = 5 and 3x + y = 7 yields a unique solution or not.

Solution:

Given Equation,

• 2x + 3y = 5
• 3x + y = 7

Comparing with a₁x + b₁y = -c₁ and a₂x + b₂y = -c₂

a₁ = 2, b₁ = 3

a₂ = 3, b₂ = 1

Using the condition 

a₁/a₂ ≠ b₁/b₂

2/3 ≠ 3/1

As the condition holds true the given equations give the unique solution.

Inconsistent Solution

We can have an Inconsistent solution to the pair of linear equations if the given pair of lines are parallel but their y-intercepts are not equal. For any two lines a₁x + b₁y = -c₁ and a₂x + b₂y = -c₂ to have Inconsistent solution the required condition is,

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

This can be further expanded as,

• a₁b₂ – a₂b₁ = 0
• b₁c₂ – b₂c₁ ≠ 0
• a₁c₂ – a₂c₁ ≠ 0

If the line follows the above condition then it has an Inconsistent Solution. In case of an inconsistent solution, the graph of two equations will be parallel, and hence no solution is derived.

Infinitely Many Solutions

We can have Infinitely Many solutions to the pair of linear equations if the given pair of lines are parallel but their y-intercepts are also multiple of each other. For any two lines a₁x + b₁y = -c₁ and a₂x + b₂y = -c₂ to have Inconsistent solution the required condition is,

a₁/a₂ = b₁/b₂ = c₁/c₂

This can be further expanded as,

• a₁b₂ – a₂b₁ = 0
• b₁c₂ – b₂c₁ = 0
• a₁c₂ – a₂c₁ = 0

If the line follows the above condition then it has Infinitely Many Solutions. In the case of Infinitely many solutions, the two graphs will be overlapping on each other thus giving infinitely many solutions.

Read More,

• Linear Equations in One Variable
• Solve the Linear Equation using Substitution Method
• Solving Linear Equations Using the Elimination Method

Cross multiplication method is one of the basic methods in mathematics that is used to solve the linear equations in two variables. It is one of the easiest to solve a pair of linear equations in two variables. 

Suppose we have a pair of linear equations in two variables, i.e. a₁x + b₁y = -c₁ and a₂x + b₂y = -c₂ then we can directly get their solution by using the cross multiplication method. This method is applied only when the condition,

b₂a₁ – b₁a₂ ≠ 0

is satisfied. Now let’s learn more about the cross multiplication method, its formula and derivation, and others in detail in this article.