Solve for x and y:\(\frac{x}a+\frac{y}b=2\),ax – by = (a2 – b2)

1.	Solve for x and y:\(\frac{x}a+\frac{y}b=2\),ax – by = (a2 – b2)
Answer» The given equations are: x/a + y/b = 2 ⇒ bx+ay/ab = 2 [Taking LCM] ⇒bx + ay = 2ab …….(i) Again, ax – by = (a² – b²) …..(ii) On multiplying (i) by b and (ii) by a, we get: b²x + bay = 2ab² ……..(iii) a²x – bay = a(a² – b²) …….(iv) On adding (iii) from (iv), we get: (b² + a²)x = 2a²b + a(a² – b²) ⇒(b² + a²)x = 2ab² + a³ – ab² ⇒(b² + a²)x = ab² + a³ ⇒(b² + a²)x = a(b² + a²) ⇒x = a(b²+ a²) (b²+ b²) = a On substituting x = a in (i), we get: ba + ay = 2ab ⇒ ay = ab ⇒ y = b Hence, the solution is x = a and y = b.

Answer»

The given equations are:

x/a + y/b = 2

⇒ bx+ay/ab = 2 [Taking LCM]

⇒bx + ay = 2ab …….(i)

Again, ax – by = (a² – b²) …..(ii)

On multiplying (i) by b and (ii) by a, we get:

b²x + bay = 2ab² ……..(iii)

a²x – bay = a(a² – b²) …….(iv)

On adding (iii) from (iv), we get:

(b² + a²)x = 2a²b + a(a² – b²)

⇒(b² + a²)x = 2ab² + a³ – ab²

⇒(b² + a²)x = ab² + a³

⇒(b² + a²)x = a(b² + a²)

⇒x = a(b²+ a²) (b²+ b²) = a

On substituting x = a in (i), we get:

ba + ay = 2ab

⇒ ay = ab

⇒ y = b

Hence, the solution is x = a and y = b.

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