Derivation of Cross Multiplication Method

We can easily derive the cross multiplication method formula for a pair of linear equations in two variables by simply solving the linear equation by making their coefficients equal. This can be understood as,

Let’s take two linear equations as

• (a)₁x + (b)₁y + (c)₁ = 0…(i)
• (a)₂x + (b)₂y + (c)₂ = 0…(ii)

For solving the above pair of linear equations we make the coefficient of y in both equations equal.

Multiplying b₂ to eq(i) and b₁ to eq(ii)

(b₂)(a)₁x + (b₂)(b)₁y + (b₂)(c)₁ = 0…(iii)

(b₁)(a)₂x + (b₁)(b)₂y + (b₁)(c)₂ = 0…(iv)

Subtracting equations (iii) and (iv) we get

(b₂a₁ – b₁a₂)x + (b₂c₁ – b₁c₂) = 0

Solving for x we get,

x = (b₁c₂ – b₂c₁)/(b₂a₁ – b₁a₂)…(a)

where, (b₂a₁ – b₁a₂) ≠ 0

Similarly solving eq (i) and eq (ii) for y we get,

y = (c₁a₂ – c₂a₁)/(b₂a₁ – b₁a₂)…(b)

where, (b₂a₁ – b₁a₂) ≠ 0

Combining eq.(a) and eq.(b) we get,

x/(b₁c₂ – b₂c₁) = y/(c₁a₂ – c₂a₁) = 1/(b₂a₁ –  b₁a₂)

This is the required cross-multiplication formula.

We can easily write this formula by using the technique discussed in the image below,

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.