CROSS MULTIPLICATION METHOD

This is one of the methods we use to solve system of linear equations.

Let us consider the following system of linear equations. 

a1x + b1y + c1  =  0

a2x + b2y + c2  =  0

We have to write the coefficients of the equations and do cross multiplication as shown below. 

We write the coefficient of y and constant term and two more columns by repeating the coefficients of x and y as follows. 

The result is given by

The solution is

Examples

Example 1 :

Solve the following system of equations using cross multiplication method :

2x + 7y - 5  =  0

-3x + 8y  =  -11

Solution:

First we have to change the given linear equations in the form a1x + b1y + c1  =  0, a2x + b2y + c2  =  0.

2x + 7y - 5  =  0

-3x + 8y + 11  =  0

x/(77 + 40)  =  y/(15 - 22)  =  1/[16 + 21]

x/117  =  y/(-7)  =  1/37

x/117  =  1/37

x  =  117/37  

y/(-7)  =  1/37

y  =  -7/37

Therefore the solution is (117/37, -7/37).

Example 2 :

Solve the following system of equations using cross multiplication method :

3x + 4y  =  24

20x - 11y  =  47

Solution:

First we have to make the given equations in the form of a1x + b1y + c1  =  0, a2x + b2y + c2  =  0.

3x + 4y - 24  =  0    ----- (1)

20x - 11y - 47  =  0 ----- (2)

x/(-188 - 264)  =  y/(-480 - (-141))  =  1/(-33 - 80)

x/(-452)  =  y/(-480+141))  =  1/(-33-80)

x/(-452)  =  y/(-339)  =  1/(-113)

x/(-452)  =  1/(-113)

x  =  (-452)/(-113)

x  =  4

y/(-339)   = 1/(-113) 

y  =  (-339)/(-113)

y  =  3

Therefore solution is (4, 3).

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