Solve the linear equations: (i) mx – ny = m2 + n2

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Solve the linear equations:

(i) mx – ny = m2 + n2
x + y = 2m
(ii) 2x/a + y/b = 2
x/a – y/b = 4

Solution:

(i) mx – ny = m2 + n2 …. (1)
x + y = 2m …. (2)
We can write it as
x = 2m – y ….. (3)
Now substitute the value of x in (1)
m (2m – y) – ny = m2 + n2
By further calculation
2m2 – my – ny = = m2 + n2
Taking out y as common
m2 – y (m + n) = n2
It can be written as
m2 – n2 – y (m + n) = 0
Expanding using formula
(m – n) (m + n) – y (m + n) = 0
Taking (m + n) as common
(m + n) [(m – n) – y] = 0
So we get
m – n – y = 0
y = m – n
From equation (3)
x = 2m – (m – n)
By further calculation
x = 2m – m + n = m + n
Hence, x = m + n and y = m – n.
(ii) 2x/a + y/b = 2 …. (1)
x/a – y/b = 4 …. (2)
Adding both the equations
3x/a = 6
So we get
x = 6a/3 = 2a
Substituting x in equation (1)
2 (2a)/ a + y/b = 2
By further calculation
4a/a + y/b = 2
So we get
4 + y/b = 2
y/b = 2 – 4 = – 2
Here
y = – 2b
Therefore, x = 2a and y = – 2b.

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