In the following two questions I am trying to use formulas to find the desired parameters. I am wondering if my formulas are correct. I believe I have done the first question correct as it is only trivial but I'm not quite sure how to complete the second.
These are the formulas I am using to complete the two questions: $bk = vr, \lambda(v-1) = r(k-1)$
A BIBD has parameters $v = 47, b = 47, r = 23$. Find $k$ and $\lambda$
Solving for $k$ in the first formula $k = \frac{vr}{b} = \frac{47*23}{47} = 23$
Then plugging that value of $k$ into the second formula and solving for $\lambda = \frac{r(k-1)}{(v-1)} = \frac{23*22}{46} = 11$
A BIBD has paramters $b = 14, k =3, \lambda = 2$. Find $v$ and $r$
I am confused how to do this since both $v$ and $r$ are used in both equations
You have two simultaneous equations in two unknowns. One common method of solution is to solve one equation for one of the unknowns and plug that into the other, giving an equation in one unknown. Solve that, then plug the value back into the first. I will not plug in the given values for $b,k,\lambda$ because it is a little clearer what is going on. You could plug them in at the start. $$bk=vr\\\lambda(v-1)=r(k-1)\\r=\frac {bk}v\\\lambda(v-1)=\frac {bk}v(k-1)\\ \lambda v^2-\lambda v=bk(k-1)$$ and you have a quadratic for $v$. The quadratic formula will give you two solutions, each of which you can plug into the first equation to find the corresponding $r$.