y = 5y + 1

-1 = 4y

y = -1/4

-2y + 10 = 3

-2(-1/4) + 10 = 3

1/2 + 10 = 3

10.5 = 3

Untrue, therefore no solution.

.

-x + 2y = 3

3x - 6y = -9

multiply the top equation by -3 on both sides

3x - 6y = -9

3x - 6y = -9

the two equations are the same, so there's no unique solution

Any pair of x and y values that satisfy 3x - 6y = -9 (or just x - 2y = -3) work.

Infinite number of solutions.

.

x + 3y = 5

5x + 6y = -2

multiply the top equation by -2 on both sides

-2x - 6y = -10

5x + 6y = -2

add the bottom equation to the top

3x + 0y = -12

5x + 6y = -2

divide the top equation by 3 on both sides

x + 0y = -4

5x + 6y = -2

Now we know x = -4

multiply the top equation by -5 on both sides

-5x + 0y = 20

5x + 6y = -2

add the top equation to the bottom

-5x + 0y = 20

0x + 6y = 18

divide the bottom equation by 6 on both sides

-5x + 0y = 20

0x + y = 3

Now we know y = 3

Answer: x = -4, y = 3