You answer stems from the signatures not being deterministic.  Since they are not deterministic

k1 being the random number and s is the message

s1*k for some k makes a signature.  Now, not being deterministic makes:

s1*k1  != s1*k2

subtract r1*x1 from both sides.

s1*k1 - r1*x1 = s1*k2 - r1*x1

ergo z1 != z2.