(4+x)(5-x)/(2+x)(7-x)=1

3 min read Jun 16, 2024

Solving the Equation: (4+x)(5-x)/(2+x)(7-x) = 1

This equation presents a challenge involving rational expressions. To solve it, we need to follow a systematic approach:

1. Simplify the Expression

Cross-multiplication: Multiply both sides of the equation by the denominator of the right side (which is 1), and by the denominator of the left side: (4+x)(5-x) = (2+x)(7-x)
Expand the products: 20 - x - x² = 14 - 5x + x²

2. Rearrange and Solve the Quadratic Equation

Move all terms to one side: 2x² - 4x - 6 = 0
Simplify by dividing by 2: x² - 2x - 3 = 0
Factor the quadratic: (x - 3)(x + 1) = 0
Solve for x: x - 3 = 0 or x + 1 = 0 x = 3 or x = -1

3. Verify the Solutions

It is crucial to verify if the obtained solutions are valid by plugging them back into the original equation.

For x = 3:

(4 + 3)(5 - 3) / (2 + 3)(7 - 3) = (7)(2) / (5)(4) = 14/20 = 7/10 ≠ 1

For x = -1:

(4 - 1)(5 + 1) / (2 - 1)(7 + 1) = (3)(6) / (1)(8) = 18/8 = 9/4 ≠ 1

Therefore, neither solution is valid. This indicates that the original equation has no solution.

Conclusion

The equation (4+x)(5-x)/(2+x)(7-x) = 1 has no solutions. This means there is no value of x that can make the equation true. This can happen when the simplification process leads to an inconsistent result after verifying the solutions.