You can also use the symbolic ring:

sage: var('x,y,z,A,B,k,i,j,m')
(x, y, z, A, B, k, i, j, m)
sage: solve([x == A*i + B*j, y == A*k + B*m, z == B*(j-m) + A*(i-k)], 
[z,i,m])
[[z == x - y, i == -(B*j - x)/A, m == -(A*k - y)/B]]


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