Step 1) Solve the first equation for #y#:
#12x + y = 38#
#12x + y - color(12x) = 38 - color(12x)#
#12x - color(12x) + y = 38 - color(12x)#
#0 + y = 38 - 12x#
#y = 38 - 12x#
Step 2) Substitute #38 - 12x# for #y# in the second equation and solve for #x#:
#4x + 5y = 22#
#4x + 5(38 - 12x) = 22#
#4x + 190 - 60x = 22#
#190 - 60x + 4x = 22#
#190 - 56x = 22#
#190 - color(red)(190) - 56x = 22 - color(red)(190)#
#0 - 56x = -168#
#-56x = -168#
#(-56x)/color(red)(-56) = -168/color(red)(-56)#
#(color(red)(cancel(color(black)(-56)))x)/cancel(color(red)(-56)) = 3#
#x = 3#
Step 3) Substitute #3# for #x# in the solution to the first equation at the end of Step 1 and calculate #y#:
#y = 38 - 12x#
#y = 38 - (12 xx 3)#
#y = 38 - 36#
#y = 2#
The solution is #x = 3# and #y = 2#