Solve the following pair of linear equations.

(i) px + qy = p - q

qx - py = p + q

(ii) ax + by = c

bx + ay = 1 + c

(iii) 

ax + by = a² + b²

(iv) (a - b) x + (a + b) y = a² - 2ab - b²

(a + b) (x + y) = a² + b²

(v) 152x - 378y = - 74

-378x + 152y = - 604

(i) px + qy = p - q … (1) qx - py = p + q … (2)

Multiplying equation (1) by p and equation (2) by q,

we obtain p²x + pqy = p² - pq … (3)

q²x - pqy = pq + q² … (4)

Adding equations (3) and (4),

we obtain p²x + q² x = p² + q²

(p² + q²) x = p² + q²

From equation (1),

we obtain p (1) + qy = p - q

qy = - q

y = - 1

(ii) ax + by = c … (1) bx + ay = 1 + c … (2)

Multiplying equation (1) by a and equation (2) by b,

we obtain a²x + aby = ac … (3)

b²x + aby = b + bc … (4)

Subtracting equation (4) from equation (3),

(a² - b²) x = ac - bc – b

From equation (1), we obtain ax + by = c

(iii)

Or, bx - ay = 0 … (1)

ax + by = a² + b² … (2)

Multiplying equation (1) and (2) by b and a respectively, we obtain b²x - aby = 0 … (3)

a²x + aby = a³ + ab² … (4)

Adding equations (3) and (4), we obtain b²x + a²x = a³ + ab²

x (b² + a²) = a (a² + b²) x = a

By using (1), we obtain b (a) - ay = 0

ab - ay = 0

ay = ab

y = b

(iv) (a - b) x + (a + b) y = a² - 2ab - b² … (1)

(a + b) (x + y) = a² + b²

(a + b) x + (a + b) y = a² + b² … (2)

Subtracting equation (2) from (1),

we obtain

(a - b) x - (a + b) x = (a² - 2ab - b²) - (a² + b²) (a - b - a - b) x = - 2ab - 2b²

- 2bx = - 2b (a + b) x = a + b

Using equation (1), we obtain

(a - b) (a + b) + (a + b) y = a² - 2ab - b²a² - b² + (a + b) y = a² - 2ab - b²

(a + b) y = - 2ab

(v) 152x - 378y = - 74 ------------(1)

-378x + 152y = -604 -------- (2)

Multiply eq (2) by 152 and equation (1) by 378

378 × 152x – 378²y = -74 × 378

-378 × 152x + 152²y = -604 × 152

Adding both the questions we get

(152² – 378²)y = -119780

-119780y = -119780

y = 1

put the value in eq 1, we get x = 2