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

#-6x + 6y = 6#

#(-6x + 6y)/color(red)(6) = 6/color(red)(6)#

#(-6x)/color(red)(6) + (6y)/color(red)(6) = 1#

#-x + y = 1#

#color(red)(x) - x + y = color(red)(x) + 1#

#0 + y = x + 1#

#y = x + 1#

Step 2) Substitute #x + 1# for #y# in the second equation and solve for #x#:

#2x + y = 19# becomes:

#2x + (x + 1) = 19#

#2x + x + 1 = 19#

#3x + 1 = 19#

#3x + 1 - color(red)(1) = 19 - color(red)(1)#

#3x + 0 = 18#

#3x = 18#

#(3x)/color(red)(3) = 18/color(red)(3)#

#x = 6#

Step 3) Substitute #6# for #x# in the solution to the first equation at the end of Step 1 and calculate #y#:

#y = x + 1# becomes:

#y = 6 + 1#

#y = 7#

The solution is: #x = 6# and #y = 7# or #(6, 7)#