2x-3y=31
3x+2y=1

solve by elimination method

So,

Let's eliminate the y's.

Multiply the first equation by 2
4x - 6y = 62

Multiply the second equation by 3
9x + 6y = 3

Add the two equations together
13x = 65

Divide both sides by 13
x = 5

Substitute in the second equation
3(5) + 2y = 1
15 + 2y = 1

Subtract 15 from both sides
2y = -14

Divide both sides by 2
y = -7

Check
3(5) + 2(-7) = 1
15 + -14 = 1
1 = 1 This checks.

2(5) - 3(-7) = 31
10 - -21 = 31
10 + 21 = 31
31 = 31 This also checks.

S = {(5,-7)}

Yes, two vertical lines can be complementary