SOLVING LINEAR EQUATIONS USING 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

Solve the following system of equations using cross multiplication method.

(i)   3x + 4y = 24, 20x - 11y = 47

(ii)  0.5x + 0.8y = 0.44 , 0.8x + 0.6y = 0.5

(iii) (3x/2) - (5y/3) = -2 , (x/3) + (y/2) = 13/6

(iv) (5/x)-(4/y)  =  -2 , (2/x)+(3/y)  =  13

Detailed Solution

(i)   3x + 4y  =  24, 20x - 11y  =  47

Solution :

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)

So, the solution is (4, 3).

(ii)  0.5x + 0.8y = 0.44 , 0.8x + 0.6y = 0.5

Solution :

0.5x+0.8y  =  0.44  ----- (1)

0.8x+0.6y  =  0.5  ----- (2)

To make the decimal numbers into integers, we have to multiply the first equation by 100 and the second equation by 10.

50x+80y-44  =  0 ----- (1)

8x+6y-5  =  0  ----- (2)

x/(-400-(-264))  =  y/(-352 -(-250))  =  1/(300-640)

x/(-400+264)   =  y/(-352+250))  =  1/(-340)

x/(-136)  =  y/(-102)  =  1/(-340)

So, the solution is (0.4, 0.3).

(iii) (3x/2) - (5y/3) = -2 , (x/3) + (y/2) = 13/6

Solution :

9x - 10y  =  -12

(2x + 3y)/6  =  13/6

9x-10y+12  =  0 ----- (1)

2x+3y-13  =  0 ----- (2)

x/(130-36)  =  y/(24 -(-117))  =  1/(27-(-20))

x/(94)  =  y/(24+117))  =  1/(27+20)

x/(94)  =  y/(141)  =  1/(47)

So, the solution is (2, 3).

(iv) (5/x)-(4/y)  =  -2 , (2/x)+(3/y)  =  13

Solution :

Let 1/x = a and 1/y = b

5a-4b  =  - 2   

2a+3 b  =  13   

5a - 4b + 2 = 0   ----- (1)

2a + 3b - 13 = 0  ----- (2)

a/(52-6)  =  b/(4 -(-65))   =  1/(15-(-8))

a/46  =  b/(4 +65)   =  1/(15+8)

a/(46)   =  b/(69)   =  1/(23)

x  =  1/2 and y  =  1/3

So, the solution is (1/2, 1/3).

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