Step 1) Solve the first equation for #y#:

#4x - y = 14#

#4x - y + color(red)(y) - color(blue)(14) = 14 + color(red)(y) - color(blue)(14)#

#4x - 0 - color(blue)(14) = 14 - color(blue)(14) + y#

#4x - 14 = 0 + y#

#4x - 14 = y#

#y = 4x - 14#

Step 2) Substitute #4x - 14# for #y# in the second equation and solve for #x#:

#7x + 3y = 10# becomes:

#7x + 3(4x - 14) = 10#

#7x + (3 xx 4x) - (3 xx 14) = 10#

#7x + 12x - 42 = 10#

#19x - 42 = 10#

#19x - 42 + color(red)(42) = 10 + color(red)(42)#

#19x - 0 = 52#

#19x = 52#

#(19x)/color(red)(19) = 52/color(red)(19)#

#(color(red)(cancel(color(black)(19)))x)/cancel(color(red)(19)) = 52/19#

#x = 52/19#

Step 3) Substitute #52/49# for #x# in the solution to the first equation at the end of Step 1 and calculate #y#:

#y = 4x - 14# becomes:

#y = (4 xx 52/19) - 14#

#y = 208/19 - (19/19 xx 14)#

#y = 208/19 - 266/19#

#y = -58/49#

The solution is: #x = 52/19# and #y = -58/49 or #(52/19, -58/49)#