Solve simultaneous linear equations: (i) 2/x + 2/3y = 1/6

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Solve simultaneous linear equations:

(i) 2/x + 2/3y = 1/6
2/x – 1/y = 1
(ii) 3/2x + 2/3y = 5
5/x – 3/y = 1.

Solution:

(i) 2/x + 2/3y = 1/6 ….. (1)
2/x – 1/y = 1 ….. (2)
By subtracting both the equations
5/3y = -5/6
By cross multiplication
– 15y = 30
By division
y = 30/ -15 = – 2
Substitute the value of y in equation (1)
2/x + 2/ (3 × (-2)) = 1/6
By further calculation
2/x – 1/3 = 1/6
So we get
2/x = 1/6 + 1/3
Taking LCM
2/x = (1 + 2)/ 6 = 3/6
By cross multiplication
x = (2 × 6)/3 = 12/3 = 4
Therefore, x = 4 and y = – 2.
(ii) 3/2x + 2/3y = 5 ….. (1)
5/x – 3/y = 1 ….. (2)
Multiply equation (1) by 1 and (2) by 2/9
3/2x + 2/3y = 5
10/9x – 2/3y = 2/9
By adding both the equations
(3/2 + 10/9)1/x = 5 + 2/9
Taking LCM
(27 + 20)/ 18 × 1/x = (45 + 2)/ 9
By further calculation
47/18x = 47/9
By cross multiplication
x = (47 × 9)/ (47 × 18) = ½
Substitute the value of x in equation (2)
5/ ½ – 3/y = 1
By further calculation
10 – 3/y = 1
3/y = 10 – 1 = 9
So we get
y = 3/9 = 1/3
Therefore, x = ½ and y = 1/3.

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