How do you solve #2/3 = 2 - (5x-3)/(x-1)#?

To solve the equation 2/3 = 2 - (5x-3)/(x-1), we can start by multiplying both sides of the equation by the common denominator, which is (x-1). This will help us eliminate the fractions.

First, multiply 2/3 by (x-1):

(2/3) * (x-1) = 2 - (5x-3)/(x-1)

Next, simplify the left side of the equation:

2(x-1)/3 = 2 - (5x-3)/(x-1)

Now, distribute the 2 on the left side:

(2x-2)/3 = 2 - (5x-3)/(x-1)

Next, multiply both sides of the equation by 3 to eliminate the fraction:

3 * (2x-2)/3 = 3 * (2 - (5x-3)/(x-1))

Simplifying further:

2x-2 = 6 - 3(5x-3)/(x-1)

Now, distribute the -3 on the right side:

2x-2 = 6 - (15x-9)/(x-1)

Next, multiply both sides of the equation by (x-1) to eliminate the fraction:

(x-1)(2x-2) = (x-1)(6 - (15x-9)/(x-1))

Expanding and simplifying:

2x^2 - 2x - 2 = 6(x-1) - 15x + 9

Now, distribute the 6 on the right side:

2x^2 - 2x - 2 = 6x - 6 - 15x + 9

Combine like terms:

2x^2 - 17x + 1 = 0

This is a quadratic equation. To solve it, you can use factoring, completing the square, or the quadratic formula.