1) Solve by factoring:
$\displaystyle (u+3)/(2u-7)=(2u-1)/(u-3)$
2) Find the exact solution to:
$\displaystyle 2(k-1)]=(4-5k)/(k+1)$
3) Solve this equation:
$\displaystyle (3t)/(t^2-6)=Sqrt3$
1) $\displaystyle \frac {u+3}{2u-7} = \frac {2u-1}{u-3}$
Cross Multiply to get:
$\displaystyle
\frac {(u + 3)(u - 3)}{2u - 7} = 2u - 1$
Multiply both sides by $\displaystyle 2u - 7$:
$\displaystyle (u + 3)(u - 3) = (2u - 7)(2u - 1)$
Now multiply out:
$\displaystyle u^2 + 3u - 3u - 9 = 4u^2 - 2u - 14u + 7$
Move everything to one side, then simplify:
$\displaystyle -3u^2 + 16u -16 = 0$
Now use the quadratic equation to find the roots.
2) $\displaystyle 2k - 2 = \frac {4 - 5k}{k + 1}$
Multiply both sides by $\displaystyle k + 1$:
(2k - 2)(k + 1) = 4 - 5k
Multiply out:
$\displaystyle
2k^2 + 2k - 2k - 2 = 4 - 5k$
Bring everything to one side, then simplify:
$\displaystyle 2k^2 + 5k - 6 = 0$
Now use the quadratic equation.
3)