Question

$x/a-y/b=a-b ax+by=a3-b3$

Answer 1

Given

• $\dfrac{x}{a} - \dfrac{y}{b} = a - b$

• $ax + by = a^{3} - b^{3}$

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To Find

• The value of 'x' and 'y'.

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Solution

Let's solve the question using elimination method.

Simplify the first equation.

⇒ $\dfrac{x}{a} - \dfrac{y}{b} = a - b$

⇒ $\dfrac{xb - ya}{ab} = a - b$

⇒ $xb- ay = ab ( a-b)$

⇒ xb - ay = a²b - ab²         - (i)

⇒ ax + by = a³ + b³      - (ii)

Now, multiply (i) by b.

⇒ b (xb) - b (ay) = b (a²b - ab²)

⇒ xb² - aby = a²b² - ab³      - (iii)

Now, multiply (ii) by a.

⇒ a (ax) + a (by) = a (a³ + b³)    

⇒ a²x + aby = a⁴ + ab³       - (iv)

Now add (iii) and (iv)

⇒ xb² - aby + a²x + aby = a²b² - ab³ + a⁴ + ab³

⇒ xb² + a²x = a²b² + a⁴

⇒ x (b² + a²) = a² (b² + a²)

⇒ x = $\dfrac{a^{2} (b^{2}+a^{2})}{b^{2}+{a^{2}}}$

⇒ x = a²

Substitute the value of 'x' in (ii)

⇒ ax + by = a³ + b³      

⇒ a (a²) + by = a³ + b³

⇒ a³ + by = a³ + b³

⇒ by = a³ + b³ - a³

⇒ by = b³

⇒ y = b³/b

⇒ y = b²

∴ x = a² and y = b².

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