Solve simultaneous linear equations: (i) 4x + (x – y)/ 8 = 17

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Solve simultaneous linear equations:

(i) 4x + (x – y)/ 8 = 17
2y + x – (5y + 2)/3 = 2
(ii) (x + 1)/2 + (y – 1)/3 = 8
(x – 1)/3 + (y + 1)/ 2 = 9.

Solution:

(i) 4x + (x – y)/ 8 = 17
2y + x – (5y + 2)/3 = 2
We can write it as
4x + (x – y)/ 8 = 17
(32 + x – y)/ 8 = 17
By further calculation
(33x – y)/ 8 = 17
By cross multiplication
33x – y = 136 ….. (1)
2y + x – (5y + 2)/3 = 2
Taking LCM
[3 (2y + x) – 5 (5y + 2)]/ 3 = 2
By further calculation
6y + 3x – 5y – 2 = 2 × 3
So we get
y + 3x – 2 = 6
3x + y = 6 + 2
3x + y = 8 ….. (2)
By adding both the equations
36x = 144
By division
x = 144/36 = 4
Substitute the value of x in equation (1)
33 × 4 – y = 136
By further calculation
132 – y = 136
– y = 136 – 132
So we get
– y = 4
y = – 4
Therefore, x = 4 and y = – 4.
(ii) (x + 1)/2 + (y – 1)/3 = 8
(x – 1)/3 + (y + 1)/ 2 = 9
We can write it as
(x + 1)/2 + (y – 1)/3 = 8
Taking LCM
(3x + 3 + 2y – 2)/6 = 8
By further calculation
3x + 2y + 1 = 48
So we get
3x + 2y = 47 ….. (1)
(x – 1)/3 + (y + 1)/ 2 = 9
Taking LCM
(2x – 2 + 3y + 3)/6 = 9
By further calculation
2x + 3y + 1 = 54
So we get
2x + 3y = 53 ….. (2)
By adding equation (1) and (2)
5x + 5y = 100
Dividing by 5
x + y = 20 …… (3)
By subtracting equation (1) and (2)
x – y = – 6 ….. (4)
Now add equation (3) and (4)
2x = 14
x = 14/2 = 7
Subtracting equation (4) and (3)
2y = 26
y = 26/2 = 13
Therefore, x = 7 and y = 13.

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