2x/(x-1)= 5/(x-3)
To solve the given equation, we'll start by getting rid of the fractions.
To do this, we can multiply both sides of the equation by the least common denominator (LCD) of the two fractions, which in this case is (x-1)(x-3).
When we multiply both sides by the LCD, it cancels out the denominators and leaves us with:
2x(x-3) = 5(x-1)
Now, let's simplify:
2x^2 - 6x = 5x - 5
To solve for x, we'll gather all the x terms on one side of the equation and all the constant terms on the other side.
Let's subtract 5x from both sides of the equation:
2x^2 - 11x = -5
Now, add 5 to both sides:
2x^2 - 11x + 5 = 0
We now have a quadratic equation. To solve it, we can either factor or use the quadratic formula.
Let's try factoring:
To factor the quadratic, we need to find two numbers whose product is 2*5 = 10 and whose sum is -11.
After some trial and error, we find that -1 and -10 satisfy these conditions. Thus, we can factor the equation as:
(2x - 1)(x - 5) = 0
Now, we can set each factor equal to zero and solve for x:
2x - 1 = 0 ---> 2x = 1 ---> x = 1/2
x - 5 = 0 ---> x = 5
So, the solutions to the equation 2x/(x-1) = 5/(x-3) are x = 1/2 and x = 5.