Strategy: There are several different options to solving these equations. This one may be susceptible to picking one of them and solving for #x# in terms of #y#. Then plug that equation with #y#s into the other equation. Then solve for #y#. Finally, with #y# equal to some value, plug it in to the first equation and solve for #x#.
Step 1. Pick one equation and solve for #x#.
Pick one of the two equations. It doesn't matter which.
#36x+19y=35#
Subtract #19y# from both sides.
#36x=-19y+35#
Divide both sides by #36#.
#color(red)(x=-19/36 y +35/36)#
Step 2. Plug that equation into the other, #10x+12y=19#.
#10color(red)(x)+12y=19#
#10(color(red)(-19/36 y +35/36))+12y=19#
Step 3. Solve for #y#.
Multiply #10# through.
#-190/36y+350/36+12y=19#
Subtract #350//36# from both sides.
#-190/36y+12y=19-350/36#
#121/18y=167/18#
Multiply both sides by #18//121#.
#y=167/18xx18/121#
#color(blue)(y=167/121)#
Step 3. Plug this into your #x# equation of Step 1.
#x=-19/36 color(blue)(y) +35/36#
#x=-19/36 (color(blue)(167/121)) +35/36#
#x=-19/36 (color(blue)(167/121)) +35/36#
#x=59/242#
ANSWER: #x=59/242# and #y=167/121#
