#### Solve simultaneous linear equations:

(i) px + qy = p – q

qx – py = p + q

(ii) x/a – y/b = 0

ax + by = a^{2} + b^{2}.

**Solution:**

(i) px + qy = p – q …. (1)

qx – py = p + q ….. (2)

Now multiply equation (1) by p and (2) by q

p^{2}x + pqy = p^{2} – pq

q^{2}x – pqy = pq + q^{2}

By adding both the equations

(p^{2} + q^{2}) x = p^{2} + q^{2}

By further calculation

x = (p^{2} + q^{2})/ (p^{2} + q^{2}) = 1

From equation (1)

p × 1 + qy = p – q

By further calculation

p – qy = p – q

So we get

qy = p – q – p = – q

Here

y = -q/q = – 1

Therefore, x = 1 and y = – 1.

(ii) x/a – y/b = 0

ax + by = a^{2} + b^{2}

We can write it as

x/a – y/b = 0

Taking LCM

(bx – ay)/ab = 0

By cross multiplication

bx – ay = 0 …… (1)

ax + by = a^{2} + b^{2} ….. (2)

Multiply equation (1) by b and equation (2) by a

b^{2}x – aby = 0

a^{2}x + aby = a^{2} + ab^{2}

By adding both the equations

(a^{2} + b^{2})x = a^{2}+ ab^{2} = a (a^{2} + b^{2})

So we get

x = a (a^{2} + b^{2})/ a^{2} + b^{2} = a

From equation (2)

b × a – ay = 0

By further calculation

ab – ay = 0

ay = ab

So we get

y = ab/a = b

Therefore, x = a and y = b.

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