450.sagews

var('Cv Vr k g SWL Design')
assume(Cv>1,Vr>=0,k>0,g>0,SWL>0,Design>0)
eqns={1:Cv==1+Vr*sqrt(k/(g*SWL)),2:SWL*Cv==Design}
eqns

{1: Cv == Vr*sqrt(k/(SWL*g)) + 1, 2: Cv*SWL == Design}

power_eq = lambda eq, n: eq.operator()(eq.lhs()^n, eq.rhs()^n) #from mhansen on sage-devel
subs_eq = lambda eq, *args: eq.operator()(eq.lhs().subs_expr(*args),eq.rhs().subs_expr(*args))

var('alpha')
assume(alpha>0)
eqns[3]=alpha==k*Vr^2/g/Design

eqns[4]=subs_eq(power_eq(eqns[1]-1,2),*solve(map(eqns.get,range(2,3+1)),k,SWL)[0]);eqns[4]

(Cv - 1)^2 == Cv*alpha

eqns[5]=(eqns[4]-alpha*Cv).expand();eqns[5]

Cv^2 - Cv*alpha - 2*Cv + 1 == 0

eqns[6]=solve(eqns[5],Cv)[1];eqns[6]

Cv == 1/2*sqrt(alpha + 4)*sqrt(alpha) + 1/2*alpha + 1

var('DF BST NLL')
eqns[7]=DF==BST*NLL/SWL;print eqns[7]
eqns[8]=DF==5;print eqns[8]
eqns[9]=DF==5*Cv/2; eqns[9]

DF == BST*NLL/SWL DF == 5 DF == 5/2*Cv

map(eqns.get,range(7,8+1))

[DF == BST*NLL/SWL, DF == 5]