Question

solve the following system of equation x+y=a+b
ax-by= a^2 +b^2

Answer 1

The answer is given below :

The given equations are

x + y = a + b .....(i)
ax - by = a² + b² .....(ii)

Now, multiplying (i) by a and (ii) by 1, we get

ax + ay = a(a + b)

⇒ ax + ay = a² + ab .....(iii)

ax - by = a² + b² .....(ii)

Now, subtracting (ii) from (iii), we get

ax + ay - ax + by = a² + ab - a² - b²

⇒ ay + by = ab - b²

⇒ (a + b)y = b(a - b)

⇒ y = b(a - b)/(a + b)

Now, putting y = b(a - b)/(a + b) in (i), we get

x + b(a - b)/(a + b) = a + b

⇒ x = (a + b) - b(a - b)/(a + b)

⇒ x = [(a + b)² - b(a - b)]/(a + b)

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

⇒ x = (a² + ab + 2b²)/(a + b)

Therefore, the required solution be

x = (a² + ab + 2b²)/(a + b),

y = b(a - b)/(a + b)

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