Question

Using properties of proportion solve for $$x$$, given $$\frac{\sqrt{5x} + \sqrt{2x - 6}}{\sqrt{5x} - \sqrt{2x - 6}} = 4$$.

Answer 1

Given equation: $$\frac{\sqrt{5x} + \sqrt{2x - 6}}{\sqrt{5x} - \sqrt{2x - 6}} = \frac{4}{1}$$

Applying componendo and dividendo:

$$\frac{\left(\sqrt{5x} + \sqrt{2x - 6}\right) + \left(\sqrt{5x} - \sqrt{2x - 6}\right)}{\left(\sqrt{5x} + \sqrt{2x - 6}\right) - \left(\sqrt{5x} - \sqrt{2x - 6}\right)} = \frac{4 + 1}{4 - 1}$$

Simplifying both sides: $$\frac{2\sqrt{5x}}{2\sqrt{2x - 6}} = \frac{5}{3}$$

$$\frac{\sqrt{5x}}{\sqrt{2x - 6}} = \frac{5}{3}$$

Squaring both sides: $$\frac{5x}{2x - 6} = \frac{25}{9}$$

Dividing both numerators by $$5$$: $$\frac{x}{2x - 6} = \frac{5}{9}$$

Cross-multiplying and solving for $$x$$:

$$9x = 5(2x - 6)$$ 

$$9x = 10x - 30$$

$$10x - 9x = 30$$

$$x = 30$$