How do you simplify #2/( x - 4) + 3/ ( x - 2) = (6x)/ ( x ^ { 2} - 6x + 8)#?

First, you need to find the least common denominator, just like when you add a fraction with no variables.
The common denominator of #(2)/(x-4)# and #(3)/(x-2)# is #(x-4)##(x-2)#.
If you factor the bottom of the right side of the equation, #x^2-6x+8#, you will also get the answer #(x-4)##(x-2)#.

Now, multiply the numerator of each fraction by the part of the LCD not already in the denominator, like so:
#(2(x-2))/((x-2)(x-4))#+#(3(x-4))/((x-2)(x-4))#
You can now combine the fractions because they have the same denominator.
#((2(x-2))+(3(x-4)))/(x^2-6x+8)#
Multiply/expand with distribution.
#(5x-16)/(x^2-6x+8)#
Multiply both sides by the denominator to remove it.
#((5x-16)/(x^2-6x+8))(x^2-6x+8)=((6x)/(x^2-6x+8))(x^2-6x+8)#

Simplify
#5x-16=6x#

Add 16 to both sides.
#5x=6x+16#

Subtract 6x from both sides.
#5x-6x=6x+16-6x#

Simplify
#-x=16#

Divide by #-1# to isolate #x#.
#(-x)/(-1)=(16)/(-1)#

Simplify

#x=-16#

#2/(x-4)+3/(x-2)=(6x)/(x^2-6x+8)#

#:.2/(x-4)+3/(x-2)=(6x)/((x-2)(x-4))#

#:.(2(x-2)+3(x-4)=6x)/((x-2)(x-4))#

multiply both sides by#(x-2)(x-4)#

#:.2x-4+3x-12=6x#

#:.2x+3x-6x=4+12#

#:.-x=16#

#:.color(magenta)(x=-16#

~~~~~~~~~~~~~~~~~~

check:-

substitute# x=-16#

#:.2/((-16)-4)+3/((-16)-2)=(6(-16))/((-16)^2-6(-16)+8)#

#:.2/-20+3/-18=-96/(256+96+8)#

#:.-0.1+(-0.166666666)=-96/360#

#:.color(magenta)(-0.266666666=-0.266666666#