x-2/(x-3)=(x-1)/(3-x)
we can make the denomenators the same by multiplyting the left side by -1/-1
-x+2/(3-x)=(x-1)/(3-x)
minus 2/(3-x) from both sides
-x=(x-1-2)/(3-x)
-x=(x-3)/(3-x)
times both sides by 3-x
-x(3-x)=(x-3)
-3x+x^2=x-3
minus x both sides and add 3
x^2-4x+3=0
factor
(x-1)(x-3)=0
set to zero
x-1=0
x=1
x-3=0
x=3
x=1 or 3