find x
4x-5/1-5x = x+1/x+7
please can you show me the method
thanks
Sure! To solve this equation for x, we'll need to follow a step-by-step method. Here's how:
Step 1: Clear the fractions by multiplying both sides of the equation by the common denominator of all the fractions, which is (x+7)(x+1):
[(4x - 5) / (1 - 5x)] * (x+7)(x+1) = (x+1) / (x+7)
Step 2: Distribute and simplify:
(4x - 5)(x+7)(x+1) = (x+1)^2
Step 3: Expand and simplify both sides of the equation:
(4x^2 + 28x - 5x - 35)(x+1) = x^2 + 2x + 1
(4x^2 + 23x - 35)(x+1) = x^2 + 2x + 1
Step 4: Expand the equation:
4x^3 + 23x^2 - 35x + 4x^2 + 23x - 35 = x^2 + 2x + 1
4x^3 + (23x^2 + 4x^2) + (23x + 23x) + (-35 - 35) = x^2 + 2x + 1
4x^3 + 27x^2 + 46x - 70 = x^2 + 2x + 1
Step 5: Move all the terms to one side of the equation:
4x^3 + 27x^2 + 46x - x^2 - 2x - 1 - 70 = 0
4x^3 + 26x^2 + 44x - 71 = 0
Step 6: Solve the cubic equation. Unfortunately, there is no general formula to solve cubic equations, but you can use numerical methods like the Rational Root Theorem, synthetic division, or a graphing calculator to estimate the solutions.
Alternatively, you can use numerical methods such as using an online equation solver or computer software to find the approximate solutions for x.
I hope this helps! Let me know if you have any further questions.