2x-3y=31

3x+2y=1

Solve by elimination method

2x - 3y = 31 ⇒ 6x - 9y = 93

3x + 2y = 1 ⇒ 6x + 4y = 2

-5y = 91

-5 - 5

y = - 18 1/5

2x - 3 (-18 1/5) = 31

2x + 54 3/5 = 31

- 54 3/5 - 54 3/5

2x = - 24 3/5

2 2

x = - 12 3/10

(x, y) = (-12 3/10, - 18 1/5)

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) }