How many solutions does this equation have 9x + 10= 1—3(5-3x)

There is no solution to that equation.

There isn't a solution. Due to the equations being equivalent, and the variable x being apart of both, you need to isolate x to one side. This is what I ended up doing:
9x+10 = 1-3(5-3x)
9x+10 = 1-15+9x
9x+10 = 9x-14 
     -10        -10
9x = 9x-24
At this point, you can see the problem. I have to subtract the 9x from the right side in order to isolate the x to the left, but by doing so, I also get rid of the x on the left, leaving me with:
0 = -24
But 0 can't equal -24. Therefore, I'm left with the conclusion that this equation is unsolvable.