#### Solve the linear equations:

(i) mx – ny = m^{2} + n^{2}

x + y = 2m

(ii) 2x/a + y/b = 2

x/a – y/b = 4

**Solution:**

(i) mx – ny = m^{2} + n^{2} …. (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 = m^{2} + n^{2}

By further calculation

2m^{2} – my – ny = = m^{2} + n^{2}

Taking out y as common

m^{2} – y (m + n) = n^{2}

It can be written as

m^{2} – n^{2} – 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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