Solving simultaneous equations in Sage

Question

In Sage (using the Sage terminal in Sage Cloud), I expected the following to yield the result [t == (1/2)]. However, it yields the result [].

sage: var('x1 y1 x2 y2 t')
(x1, y1, x2, y2, t)
sage: eq1 = x1==t
sage: eq2 = y1==t
sage: eq3 = x2==t
sage: eq4 = y2==1-t
sage: solve([eq1,eq2,eq3,eq4,x1==x2,y1==y2],t)
[]

The following reformulation doesn't help:

sage: eq5 = x1==x2
sage: eq6 = y1==y2
sage: solve([eq1,eq2,eq3,eq4,eq5,eq6],t)
[]

Where am I going wrong?

Answer 1

If you write

solve([eq1,eq2,eq3,eq4,x1==x2,y1==y2],t)

Then t is the variable you are solving for and x1, x2, y1, y2 are free parameters. There is obviously no solution which is valid for any value of those parameter. However, if you ask:

solve([eq1,eq2,eq3,eq4,x1==x2,y1==y2],t,x1,x2,y1,y2)

Then you get what you expect:

[[t == (1/2), x1 == (1/2), x2 == (1/2), y1 == (1/2), y2 == (1/2)]]