Hello minimath
I'll try my best to help you but english is my second language so I'm sorry for the mistakes

First to solve this problem we have to understand that because all the four vectors are equilebrate their total will be equal to zero. To give you an image, think of two people pulling on a rope to make the other felt, in fact as long as no one move we know that both use the same strengh on the rope but in opposite way so the total strengh is zero. (like in your problem)

The diffenrence in my problem and yours, mine just include vector parallel to the ground (or ''X's'' axis) but you have to mind vertical vector's too.

second of all, you have to understand vector can be split into two constituent, x and y.

So, let's try it!

horizontally:
Pcos0 + Qcos70 + 8cos195 + 5cos270=0
(cos270=0)-> Pcos0 + Qcos70 + 8cos195=0
(cos0=1)-> P+ Qcos70 + 8cos195=0

vertically:
Psin0 + Qsin70 + 8sin195 + 5sin270 = 0
(sin270=-1)-> Psin0 + Qsin70 + 8sin195 -5= 0
(sin0=0)->  Qsin70 + 8sin195 -5= 0

we last with those two equation:
P+ Qcos70 + 8cos195=0
Qsin70 + 8sin195 -5= 0

we can see that there is two unknown variable in the horizontally equation, the P and de Q
In fact it is impossible to solve an equation with two variables.
The second equation has only one unknown variable, the Q
so if we isolate it we got:

1) Qsin70 + 8sin195 -5= 0
2) Qsin70 + 8sin195 =5
3) Qsin70 = 5-8sin195
4) Q5-8sin195)/ sin70 ≈7,52

We just found the Q, now the P
we take the other equation and we introduce the knew value

1) P+ Qcos70 + 8cos195=0
2) P= -Qcos70 - 8cos195
3) Q=7,52
4) P=-7,52cos70 -8cos195≈5,15

so,
Q=7,52
P=5,15

Hope I've been useful, samuelbm