In order to solve for three unknowns you need three equations...
I tried my best or may be I tried it the wrong way as linear algebra is not the topic I know
I need to solve for p, q, and r in the following linear equation, where each of these variables represents a probability measure whose sum is 1
Thank you in advance
There are an infinite number of solutions. This can be shown by assuming that r is know and, since it is a probability, falls withing certain limits (in this particular case between 1/6 and 1/2 inclusive because p and q must be between 0 and 1 inclusive) and solve the equation for p and q in terms of r.
You could, for example, solve for, say, p and q in terms of r. Multiply the second equation by 2 to get 2p+ 2q+ 2r= 2. Now subtract that from the first equation, eliminating q: 2p- r= 0.5 so that p= r/2+ 1/4.
I have nothing to add to what has already been said about the problem AS POSTED.
I suggest that you explain in detail the problem from which you derived the other two equations. There may be sufficient information buried in the problem to extract a third equation.