Solve simultaneous linear equations:
(i) 3x + 14y = 5xy
21y – x = 2xy
(ii) 3x + 5y = 4xy
2y – x = xy.
Solution:
(i) 3x + 14y = 5xy
21y – x = 2xy
Now dividing each equation by xy of x ≠ 0, y ≠ 0
3x/xy + 14y/xy = 5xy/xy
By further calculation
3/y = 14/x = 5 ….. (1)
(ii) 3x + 5y = 4x
2y – x = xy
We can write it as
3x + 5y = 4xy
– x + 2y = xy
Divide each equation by xy if x≠ 0 and y ≠ 0
3x/xy + 5y/xy = 4xy/xy
So we get
3/y + 5/x = 4 ….. (1)
-x/xy + 2y/xy = xy/xy
So we get
-1/y + 2/x = 1 ….. (2)
Now multiply equation (1) by 1 and (2) by 3
3/y + 5/x = 4
-3/y + 6/x = 3
By adding both the equations
11/x = 7
So we get
x = 11/7
Substitute the value of x in equation (2)
-1/y + 2/11/7 = 1
By further calculation
-1/y + (2 × 7)/ 11 = 1
-1/y + 14/11 = 1
We can write it as
-1/y = 1 – 14/11
Taking LCM
-1/y = (11 – 14)/ 11
So we get
-1/y = -3/11
By cross multiplication
-3y = – 11
y = – 11/-3 = 11/3
Therefore, x = 11/7 and y = 11/3.
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