Answer:
1) 5*x + 6 = 2 + 3*x
5*x - 3*x = 2 - 6
(5 - 3)*x = -4
2*x = - 4
x = -4/2 = -2
x = -2
We have only one solution.
2) 2*(6 - 2*y) = -1*(4*y - 9)
12 - 4*y = -4*y + 9
12 - 9 = -4*y + 4*y = 0
3 = 0
This is absurd, so this equation has no solution.
3) 2*z - 6 = 2*(z + 2) - 10
2*z - 6 = 2*z + 4 - 10
2*z - 6 = 2*z - 6
Here we have the exact same thing on both sides of the equation, then we have infinite solutions, this happens because z can take any value, and the equation will be true always.
4) We want to have no solutions, so we need to end with something like:
1 = 6
so, we start with:
5*x + 1 = 5*x + A
We want to find the value of A.
First, we can subtract 5*x in both sides, so we get:
1 = A
Now we just need to take A different than 1, for example, if A = 2, then:
1 = 2
In this case the equation has no solutions, then we have:
5*x + 1 = 5*x + 2
5) We want to have one solution:
3*x - 3 = A*x + 11
We want to end with something like x = k
Let's solve the equation for x:
3*x - 3 = A*x + 11
3*x - A*x = 11 + 3
(3 - A)*x = 14
if we take A = 2, then:
(3 - 2)*x = 14
x = 14
So this equation has only one solution, the equation is:
3*x - 3 = 2*x + 11
6) We want to have infinitely many solutions, then we need to have the same thing on both sides, then we need to have an equation like:
8*x - 7 = 8*x - 7
since it is exactly in the halfway between 
.