Why does Sage not solve this system of (linear) equations, but then it does after some manual substitutions? - sage

Question

I have a system of equations in which there are two unknowns. Originally (the system is derived from the analysis of an electrical circuit) I have four equations with four unknowns, but two of them (irc1 and ire) are linear combinations of the "real" unknowns (ib1 and ib2). All the other variables are treated as parameters of the circuit and its devices. When I try to solve the full set of four equations, Sage does not give an answer. This:
ib1, ib2, ire, irc1, rfb, re, vcc, veb1, veb2, rc1, b1, b2 = var('ib1 ib2 ire irc1 rfb re vcc veb1 veb2 rc1 b1 b2')
solve([ veb1 == -ib1*rfb + ire*re,
vcc == ire*re + veb2 + rc1*irc1,
irc1 == b1*ib1 + ib2,
ire == (b2+1)*ib2 - ib1
], ib1, ib2)
Yields:
[]
But if I manually substitute the last two equations into the first ones:
ib1, ib2, rfb, re, vcc, veb1, veb2, rc1, b1, b2 = var('ib1 ib2 rfb re vcc veb1 veb2 rc1 b1 b2')
solve([ veb1 == -ib1*rfb + ((b2+1)*ib2 - ib1)*re,
vcc == ((b2+1)*ib2 - ib1)*re + veb2 + rc1*(b1*ib1 + ib2)
], ib1, ib2)
Then it does output the answer I am looking for:
[[ib1 == ((b2*(vcc - veb2) + vcc - veb2)*re - ((b2 + 1)*re + rc1)*veb1)/((b1*b2*rc1 + (b1 + 1)*rc1)*re + ((b2 + 1)*re + rc1)*rfb), ib2 == (re*(vcc - veb2) + rfb*(vcc - veb2) + (b1*rc1 - re)*veb1)/((b1*b2*rc1 + (b1 + 1)*rc1)*re + ((b2 + 1)*re + rc1)*rfb)]]
I am probably missing something obvious, but I'd be grateful if someone could explain why does Sage behave like this. You can try the first example in SageMathCell using this link, and the second one (the one that provides the answer) using this link.
Also, the first example does provide an answer if the two "fake" uknowns are included in the list of variables to solve for. This:
ib1, ib2, ire, irc1, rfb, re, vcc, veb1, veb2, rc1, b1, b2 = var('ib1 ib2 ire irc1 rfb re vcc veb1 veb2 rc1 b1 b2')
solve([ veb1 == -ib1*rfb + ire*re,
vcc == ire*re + veb2 + rc1*irc1,
irc1 == b1*ib1 + ib2,
ire == (b2+1)*ib2 - ib1
], ib1, ib2, irc1, ire)
Yields:
[[ib1 == (((b2 + 1)*vcc - (b2 + 1)*veb2)*re - ((b2 + 1)*re + rc1)*veb1)/((b1*(b2 + 1) + 1)*rc1*re + ((b2 + 1)*re + rc1)*rfb), ib2 == (re*(vcc - veb2) + rfb*(vcc - veb2) + (b1*rc1 - re)*veb1)/((b1*(b2 + 1) + 1)*rc1*re + ((b2 + 1)*re + rc1)*rfb), irc1 == -((b1*(b2 + 1) + 1)*re*veb1 - ((b1*(b2 + 1) + 1)*vcc - (b1*(b2 + 1) + 1)*veb2)*re - rfb*(vcc - veb2))/((b1*(b2 + 1) + 1)*rc1*re + ((b2 + 1)*re + rc1)*rfb), ire == ((b1*(b2 + 1) + 1)*rc1*veb1 + ((b2 + 1)*vcc - (b2 + 1)*veb2)*rfb)/((b1*(b2 + 1) + 1)*rc1*re + ((b2 + 1)*re + rc1)*rfb)]]
So I could happily live with this version (link to SageMathCell) and just discard the solutions for ire and irc1, but I'd like to understand Sage's behaviour and to know what I am missing or doing wrong.
Thank you in advance.