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
(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)
x/(-452) = 1/(-113) x = (-452)/(-113) x = 4 |
y/(-339) = 1/(-113) y = (-339)/(-113) y = 3 |
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)
x/(-136) = 1/(-340) x = (-136)/(-340) x = 0.4 |
y/(-102) = 1/(-340) y = (-102)/(-340) y = 0.3 |
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)
x/(94) = 1/(47) x = 94/47 x = 2 |
y/(141) = 1/(47) y = (141)/(47) y = 3 |
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)
a/(46) = 1/(23) a = 46/23 a = 2 |
b/(69) = 1/(23) b = 69/23 b = 3 |
x = 1/2 and y = 1/3
So, the solution is (1/2, 1/3).
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