First, multiply each side of the equation by #color(red)(28)# to eliminate the fractions while keeping the equation balanced. #color(red)(28)# was chosen because it is the Lowest Common Multiply of the two fractions:
#color(red)(28)((m - 17)/4 - (6m - 7)/7) = color(red)(28) xx 1#
#(color(red)(28) xx (m - 17)/4) - (color(red)(28) xx (6m - 7)/7) = 28#
#(cancel(color(red)(28))7 xx (m - 17)/color(red)(cancel(color(black)(4)))) - (cancel(color(red)(28))4 xx (6m - 7)/color(red)(cancel(color(black)(7)))) = 28#
#7(m - 17) - 4(6 m- 7) = 28#
#(7 xx m) - (7 xx 17) - (4 xx 6m) + (4 xx 7) = 28#
#7m - 119 - 24m + 28 = 28#
#7m - 119 - 24m + 28 - color(red)(28) = 28 - color(red)(28)#
#7m - 119 - 24m + 0 = 0#
#7m - 119 - 24m = 0#
#7m - 24m - 119 = 0#
#(7 - 24)m - 119 = 0#
#-17m - 119 = 0#
#-17m - 119 + color(red)(119) = 0 + color(red)(119)#
#-17m - 0 = 119#
#-17m = 119#
#(color(red)(cancel(color(black)(-17)))m)/cancel(color(red)(-17)) = -7#
#m = -7#