Final answer:
To solve 5x/(x-2) = 7 + 10/(x-2), we first combine like terms and then cross multiply to remove the denominator, resulting in the equation 2x = 4, thus x = 2. However, x = 2 makes the original denominator zero, so the equation has no solution.
Step-by-step explanation:
Solving the Rational Equation
To solve the rational equation 5x / (x-2) = 7 + 10/(x-2), we first want to simplify the algebra by eliminating terms wherever possible. We can see that both terms on the left and the right have a common denominator of (x-2). This allows us to combine them into a single fraction on each side of the equation.
After combining we get: 5x/(x-2) - 10/(x-2) = 7. Simplify further: (5x - 10)/(x - 2) = 7. Now we can cross multiply to get a linear equation: 5x - 10 = 7(x - 2). This simplifies to 5x - 10 = 7x - 14. By moving variables to one side and constants to the other, we get: 2x = 4, and therefore, x = 2.
However, we check the answer and notice that x = 2 is not permissible because it would make the denominator zero in the original equation. Therefore, this equation has no solution.