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