Question

If $$\frac{5x + 6y}{5x - 6y} = \frac{5u + 6v}{5u - 6v}$$ show that $$\frac{x}{y} = \frac{u}{v}$$.

[Hint : We have $$\frac{5x + 6y}{5x - 6y} = \frac{5u + 6v}{5u - 6v}$$. Apply Componendo & Dividendo.]

Answer 1

Given equation: $$\frac{5x + 6y}{5x - 6y} = \frac{5u + 6v}{5u - 6v}$$

Applying componendo and dividendo on both sides:

$$\frac{(5x + 6y) + (5x - 6y)}{(5x + 6y) - (5x - 6y)} = \frac{(5u + 6v) + (5u - 6v)}{(5u + 6v) - (5u - 6v)}$$

Simplifying the numerators and denominators: $$\frac{10x}{12y} = \frac{10u}{12v}$$

Multiplying both sides by $$\frac{12}{10}$$, we get: $$\frac{x}{y} = \frac{u}{v}$$

Hence proved.