6/x-7x/5=x/10
To solve the equation 6/x - 7x/5 = x/10, we can follow these steps:
1. Simplify the fractions:
- To simplify 6/x, we keep it as it is.
- To simplify 7x/5, multiply both the numerator and denominator by 2 to eliminate the fraction: (7x * 2) / (5 * 2) = 14x/10.
- Now we have the equation: 6/x - 14x/10 = x/10.
2. Get rid of the denominators:
- To get rid of the fractions, we can multiply the whole equation by 10x (the least common multiple of 10 and x). This will cancel out the denominators.
- Multiply both sides of the equation by 10x: 10x * (6/x - 14x/10) = 10x * (x/10).
- Simplify each term: 60 - 14x^2 = x^2.
3. Rearrange the equation:
- Move all the terms to one side of the equation to set it equal to zero: x^2 + 14x^2 - 60 = 0.
- Combine like terms: 15x^2 - 60 = 0.
4. Solve the quadratic equation:
- To solve the quadratic equation, we can either factor it or use the quadratic formula. In this case, let's solve it by factoring.
- Factor out the greatest common factor, which is 15: 15(x^2 - 4) = 0.
- Set each factor equal to zero and solve for x:
- First factor: 15 = 0 (not a valid solution)
- Second factor: x^2 - 4 = 0
- Take the square root of both sides: √(x^2 - 4) = √0.
- Simplify: x^2 - 4 = 0.
- Positive square root: x^2 = 4.
- Take the square root of both sides: x = ±√4.
- Simplify: x = ±2.
So, the solutions to the equation are x = 2 and x = -2.