Solve simultaneous linear equations: (i) (7x – 2y)/ xy = 5

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Solve simultaneous linear equations:

(i) (7x – 2y)/ xy = 5
(8x + 7y)/ xy = 15
(ii) 99x + 101y = 499xy
101x + 99y = 501xy.

Solution:

(i) (7x – 2y)/ xy = 5
(8x + 7y)/ xy = 15
We can write it as
7x/xy – 2y/xy = 5
8x/xy + 7y/xy = 15
By further simplification
7/y – 2/x = 5 …. (1)
8/y + 7/x = 15 ….. (2)
Now multiply equation (1) by 7 and (2) by 2
49/y – 14/x = 35
16/y + 14/x = 30
By adding both the equations
65/y = 65
So we get
y = 65/65 = 1
Substitute the value of y in equation (1)
7/1 – 2/x = 5
By further calculation
2/x = 7 – 5 = 2
So we get
x = 2/2 = 1
Therefore, x = 1 and y = 1.
(ii) 99x + 101y = 499xy
101x + 99y = 501xy
Now divide each term by xy
99x/xy + 101y/xy = 499xy/xy
101y/xy + 99x/xy = 501xy/xy
By further calculation
99/y + 101/x = 499 ….. (1)
101/y + 99/x = 501 ….. (2)
By adding both the equations
200/y + 200/x = 1000
Divide by 200
1/y + 1/x = 5 …… (3)
Subtracting both the equations
-2/y + 2/x = – 2
Divide by 2
-1/y + 1/x = – 1 …. (4)
By adding equation (3) and (4)
2/x = 4
So we get
x = 2/4 = ½
By subtracting equation (3) and (4)
2/y = 6
So we get
y = 2/6 = 1/3
Therefore, x = ½ and y = 1/3 if x ≠ 0, y ≠ 0.

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