CLASS-9METHOD OF CROSS MULTIPLICATIONS - SIMULTANEOUS LINEAR EQUATIONS

Method of Cross Multiplication (Simultaneous Linear Equations)

Theorem – consider the system of linear equations

a₁            b₁

a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0, where -------- ≠ --------

a₂            b₂

The above system has a unique solution, given by

(b₁c₂ - b₂c₁)                   (c₁a₂ - c₂a₁)

x = ---------------, and  y = ----------------

(a₁b₂ - a₂b₁)                   (a₁b₂ - a₂b₁)

PROOF.)  The given equations are –

a₁x + b₁y + c₁ = 0 ………………...(i)

a₂x + b₂y + c₂ = 0 ………………...(ii)

multiplying (i) by b₂, (ii) by b₁ and subtracting, we get –

(a₁b₂ - a₂b₁) x = (b₁c₂ - b₂c₁)

(b₁c₂ - b₂c₁)

x = ----------------

(a₁b₂ - a₂b₁)

Multiplying (ii) by a₁, (i) by a₂ and subtracting, we get –

(a₁b₂ - a₂b₁) y = (c₁a₂ - c₂a₁)

(c₁a₂ - c₂a₁)

So, y  =  -----------------

(a₁b₂ - a₂b₁)

Thus the given system of equations has a unique solution given by –

(b₁c₂ - b₂c₁)                     (c₁a₂ - c₂a₁)

x = ----------------,  and  y = ----------------

(a₁b₂ - a₂b₁)                     (a₁b₂ - a₂b₁)

Note.– The above result may be written as

x                       y                      1

--------------- = ---------------- = ----------------

(b₁c₂ - b₂c₁)          (c₁a₂ - c₂a₁)           (a₁b₂ - a₂b₁)

Remarks: The diagram given below helps in remembering

Rule:- Numbers with downward arrows are multiplied first, and from this product, the product of numbers with upward arrows is subtracted.

x                       y                       1

---------------- = ---------------- = ----------------

(b₁c₂ - b₂c₁)            (c₁a₂ - c₂a₁)           (a₁b₂ - a₂b₁)