What you show is NOT the answer. It is AN answer.
3 equations with 4 variables is called an underdetermined problem. There is effectively no solution. If you are willing to parameterize the problem, leaving one of the variables an unknown, then you can solve 3 equations in 3 variables, with the answer as a function of the 4th variable.
That seems to be what you are asking for. And the solution is simple. Just use solve. Pick the three variables you wish to solve the problem for, and the last variable will be left as a parameter.