Maths help?

15x - 6y = 66
4x + 6y = 10________ADD

19x = 76
x = 4

16 + 6y = 10
6y = - 6
y = - 1

x = 4 , y = - 1

You seem to be expecting two different solutions for x and for y, but there is only one pair of solutions - these are simultaneous equations. 5x - 2y = 22 ....(1) 2x + 3y = 5 .....(2) multiply (1) by 3 and (2) by 2 and you have 15x - 6y = 66 ....(3) and 4x + 6y = 10 ....(4) Now add (3) & (4) to get 19x = 76 so divide both sides by 19 and x = 4. Now substitute that value for x in (2) and 8 + 3y = 5 so 3y = - 3; divide by 3 and y = -1. Now use those two values in (1) to make sure they work.

5x - 2y = 22
2x + 3y = 5

15x - 6y = 66
4x + 6y = 10
19x = 76
x = 4

2 * 4 + 3y = 5
8 + 3y = 5
3y = -3
y = -1

(4, -1)

Methodology.
1. Multiply the first equation by 3.
2. Multiply the second equation by 2.
3. Add the equations together to eliminate y and solve for x.
4. Substitute the calculated value for x into either equation to solve for y.

15x - 6y = 66
4x + 6y = 10
- - - - - - - - - -
19x = 76
x = 4

Let's plug that (x = 4) into equation #2 (although it would work to plug it into equation #1, too.
(2)(4) + 3y = 5
3y = -3
y = -1

So x = 4 and y = -1

5x - 2y = 22
2x + 3y = 5

I'll use substitution.
5x - 22 = 2y which means y = (5/2)x - 11
Put that into 2x + 3y = 5

[ 2x + 3((5/2)x - 11) = 5 ] * 2 multiple everything by 2 so I deal with whole numbers.
4x + 3(5x - 22) = 10
4x + 15x - 66 = 10
19x = 76
x = 76/19 = 4

X = 4
y = (5/2)x - 11 = (5/2)*4 - 11 = 10 - 11 = -1

X=4, Y = -1

You can check it against the original.