Solve for x and y: a2 x + b2 y = c2 , b2 x + a2 y = d2

1.	Solve for x and y: a2 x + b2 y = c2 , b2 x + a2 y = d2
Answer» The given equations are a² x + b² y = c² ………(i) b² x + a² y = d² ………(ii) Multiplying (i) by a² and (ii) by b² and subtracting, we get a⁴ x – b⁴ x = a² c² - b² d² ⇒ x = \(\frac{a^2c^2-b^2d^2}{a^4-b^4}\) Now, multiplying (i) by b² and (ii) by a² and subtracting, we get b⁴ y – a⁴ y = b² c² - a² d² ⇒ y =\(\frac{b^2c^2-a^2d^2}{a^4-b^4}\) Hence, x = \(\frac{a^2c^2-b^2d^2}{a^4-b^4}\) and y = \(\frac{b^2c^2-a^2d^2}{a^4-b^4}\)

Answer»

The given equations are

a² x + b² y = c² ………(i)

b² x + a² y = d² ………(ii)

Multiplying (i) by a² and (ii) by b² and subtracting, we get

a⁴ x – b⁴ x = a² c² - b² d²

⇒ x = \(\frac{a^2c^2-b^2d^2}{a^4-b^4}\)

Now, multiplying (i) by b² and (ii) by a² and subtracting, we get

b⁴ y – a⁴ y = b² c² - a² d²

⇒ y =\(\frac{b^2c^2-a^2d^2}{a^4-b^4}\)

Hence, x = \(\frac{a^2c^2-b^2d^2}{a^4-b^4}\) and y = \(\frac{b^2c^2-a^2d^2}{a^4-b^4}\)

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