(x+3)/(x-3)+(x)/(x-5)=(x+5)/(x-5)

3 min read Jun 16, 2024

Solving the Equation: (x+3)/(x-3) + (x)/(x-5) = (x+5)/(x-5)

This equation involves rational expressions, which means fractions with variables in the numerator and denominator. To solve it, we need to follow these steps:

1. Find a Common Denominator

The least common denominator (LCD) for all the fractions is (x-3)(x-5). We need to multiply each fraction by a suitable form of 1 to achieve this LCD:

For (x+3)/(x-3): Multiply by (x-5)/(x-5)
For (x)/(x-5): Multiply by (x-3)/(x-3)
For (x+5)/(x-5): Already has the LCD

This gives us:

[(x+3)(x-5)]/[(x-3)(x-5)] + [x(x-3)]/[(x-3)(x-5)] = [(x+5)(x-3)]/[(x-3)(x-5)]

2. Combine the Numerators

Since all the fractions now have the same denominator, we can combine their numerators:

(x+3)(x-5) + x(x-3) = (x+5)(x-3)

3. Expand and Simplify

Expand the products on both sides of the equation:

x² - 2x - 15 + x² - 3x = x² + 2x - 15

Combine like terms:

2x² - 5x - 15 = x² + 2x - 15

4. Solve for x

Move all terms to one side of the equation:

x² - 7x = 0

Factor out an x:

x(x - 7) = 0

Set each factor equal to zero and solve for x:

x = 0 or x - 7 = 0

Therefore, the solutions to the equation are:

x = 0 or x = 7

5. Check for Extraneous Solutions

It is important to check if our solutions make the original equation undefined (division by zero). Neither x = 0 nor x = 7 make any of the denominators zero. Therefore, both solutions are valid.

Conclusion:

The solutions to the equation (x+3)/(x-3) + (x)/(x-5) = (x+5)/(x-5) are x = 0 and x = 7.