$\displaystyle \frac{4m}{m-2}-\frac{13}{3m-6}=\frac{1}{3}$
I'm having problems finding a common denominator
(4 m)/(m-2)-13/(m-6) = 1/3
Multiply both sides by m-6:
(4 (m-6) m)/(m-2)-13 = (m-6)/3
Expand out terms on both sides:
(4 m^2)/(m-2)-(24 m)/(m-2)-13 = m/3-2
Write the left hand side as a single fraction:
(4 m^2-37 m+26)/(m-2) = m/3-2
Multiply both sides by m-2:
4 m^2-37 m+26 = 1/3 (m-2) m-2 (m-2)
Expand out terms on the right hand side:
4 m^2-37 m+26 = m^2/3-(8 m)/3+4
Subtract (m^2/3-(8 m)/3+4) from both sides:
(11 m^2)/3-(103 m)/3+22 = 0
Solve the quadratic equation by completing the square:
Divide both sides by 11/3:
m^2-(103 m)/11+6 = 0
Subtract 6 from both sides:
m^2-(103 m)/11 = -6
Add 10609/484 to both sides:
m^2-(103 m)/11+10609/484 = 7705/484
Factor the left hand side:
(m-103/22)^2 = 7705/484
Take the square root of both sides:
|m-103/22| = sqrt(7705)/22
Eliminate the absolute value:
m-103/22 = -sqrt(7705)/22 or m-103/22 = sqrt(7705)/22
Add 103/22 to both sides:
m = 1/22 (103-sqrt(7705)) or m-103/22 = sqrt(7705)/22
Add 103/22 to both sides:
m = 1/22 (103-sqrt(7705)) or m = 1/22 (103+sqrt(7705))
note that $\displaystyle 3m-6 = 3(m-2)$
$\displaystyle \frac{3 \cdot 4m}{3(m-2)}-\frac{13}{3(m-2)}=\frac{1(m-2)}{3(m-2)}$
denominators are all the same ... the numerator forms the equation
$\displaystyle 12m - 13 = m-2$
solve for $\displaystyle m$ , remember that m cannot equal $\displaystyle 2$.
$\displaystyle \frac{4m}{m-2}-\frac{13}{3m-6}=\frac{1}{3} $
$\displaystyle \frac{3(4m)}{3(m-2)}-\frac{13}{3(m-2)}=\frac{1}{3} $
$\displaystyle \frac{12m-13}{3(m-2)}=\frac{1}{3} $
$\displaystyle \frac{12m-13}{m-2}=1 $
$\displaystyle 12m-13 = m-2 $
$\displaystyle 11m=11$
$\displaystyle m=1$