Soving for resistor values and source current-image included

Method is correct but the current in R2 is not 4A.

 

Actually there is a way to find I_s without working out R2 or R3 first.

Have you studied KCL yet?

 

Yes. The current going into a node equals the current coming out. Google Kirchoff's Current Law.

4A leaves the node at the top of R2 so 4A must enter. 1A comes from the R3 branch so 3A must come from the R2 branch.

You can also use KCL to calculate Is.

 

Despite what I said above I find it's a bit dangerous to do "the sum going in = the sum sum out". It's better to define current going into a node as +ve and then make a sum to zero..

In other words I prefer to write..
+I_R2 +1 +(-4) = 0
then solve for I_R2

rather than write
4 = 1 + I_R2

The reason is that later on when you get more complicated circuits it's very easy to make sign errors. In many cases you don't know the direction of one or more currents. In those cases you have to define a direction as positive, then later when you solve the simultaneous equations you might discover that one of them is negative. At that point you can refer back to your definition to work out what that means.

Edit: I should add that this is my personal preference. Either way works but I find this way I make fewer sign errors.