First, multiply both sides for the equation by #color(red)(28)# to eliminate the fractions while keeping the equation balanced:

#color(red)(28)((6x - 7)/4 + (3x - 5)/7) = color(red)(28) xx (5x + 7)/28#

#(color(red)(28) xx (6x - 7)/4) + (color(red)(28) xx (3x - 5)/7) = cancel(color(red)(28)) xx (5x + 7)/color(red)(cancel(color(black)(28)))#

#(cancel(color(red)(28))7 xx (6x - 7)/color(red)(cancel(color(black)(4)))) + (cancel(color(red)(28))4 xx (3x - 5)/color(red)(cancel(color(black)(7)))) = 5x + 7#

#7(6x - 7) + 4(3x - 5) = 5x + 7#

#42x - 49 + 12x - 20 = 5x + 7#

#42x + 12x - 49 - 20 = 5x + 7#

#54x - 69 = 5x + 7#

Next, subtract #color(red)(5x)# and add #color(blue)(69)# to each side of the equation to isolate the #x# term while keeping the equation balanced:

#54x - 69 + color(blue)(69) - color(red)(5x) = 5x + 7 + color(blue)(69) - color(red)(5x)#

#54x - color(red)(5x) - 69 + color(blue)(69) = 5x - color(red)(5x) + 7 + color(blue)(69)#

#54x - 5x - 0 = 0 + 7 + 69#

#49x = 76#

Now, divide each side of the equation by #color(red)(49)# to solve for #x# while keeping the equation balanced:

#(49x)/color(red)(49) = 76/color(red)(49)#

#(color(red)(cancel(color(black)(49)))x)/cancel(color(red)(49)) = 76/49#

#x = 76/49#