Can I use regression to solve a multiple equation problem
Yes, linear regression is perfect for this.

If you knew that all the quantities were measured with perfect accuracy (no errors), you could solve using linear algebra; the system amounts to solving $Ax=y$, where $A$ is a matrix where each row corresponds to the number of each task for one person, and $x$ is a vector that reflects the time of each task, and $y$ is a vector that records the number of days each person worked.

If you suspect there might be small errors in the measurement of $y$, then using linear regression is appropriate.

You'll need that the number of people exceeds the number of tasks.

If you have a different error model, the optimal procedure might be something other than linear regression.  For instance, if you have errors in both the measurements of $y$ and $x$, PCA (also called orthogonal regression) might be more suitable.  If you have a more complex error model, you might want to ask on Stats.SE for the most appropriate inference procedure, or consider using stochastic gradient descent to minimize an appropriate loss function chosen based on your error model.