Question 251721
{{{(3t-5)/(t-1)=2+(2t)/(1-t)}}} Start with the given equation.



{{{(3t-5)/(t-1)=2+(2t)/(-1(-1+t))}}} Factor out a -1 from {{{1-t}}} to get {{{-1(-1+t)}}}



{{{(3t-5)/(t-1)=2+(2t)/(-1(t-1))}}} Rearrange the terms.



{{{(3t-5)/(t-1)=2-(2t)/(t-1)}}} Reduce.



{{{cross((t-1))((3t-5)/cross((t-1)))=2(t-1)-cross((t-1))((2t)/cross((t-1)))}}} Multiply EVERY term by the LCD {{{t-1}}} to clear out the fractions.



{{{3t-5=2(t-1)-2t}}} Simplify.



{{{3t-5=2t-2-2t}}} Distribute.



{{{3t-5=-2}}} Combine like terms on the right side.



{{{3t=-2+5}}} Add {{{5}}} to both sides.



{{{3t=3}}} Combine like terms on the right side.



{{{t=(3)/(3)}}} Divide both sides by {{{3}}} to isolate {{{t}}}.



{{{t=1}}} Reduce.



So the <i>possible</i> solution is {{{t=1}}}, but we need to check it.



Check:



{{{(3t-5)/(t-1)=2+(2t)/(1-t)}}} Start with the given equation.



{{{(3(1)-5)/(1-1)=2+(2(1))/(1-1)}}} Plug in {{{t=1}}}



{{{(3-5)/(1-1)=2+(2)/(1-1)}}} Multiply



{{{-2/0=2+2/0}}} Subtract



Since you CANNOT divide by zero, this means that {{{t=1}}} is NOT in the domain. So {{{t=1}}} can't be a solution.



=======================================================

Answer:



So there are no solutions to the equation {{{(3t-5)/(t-1)=2+(2t)/(1-t)}}}