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This system of equations is based on electronic circuit. Seems 'solve' works fine if asked to find all unknown values: ('beta' and 'i_in' is excluded as they are known)

reset()
var('i_B10 i_B6 i_B11 i_C10 i_C6 i_C11 beta i_in i_out i_E10 i_E6 i_E11')
i=solve([
  i_C10 == i_B10 * beta,
  i_E10 == i_C10 + i_B10,

  i_C6 == i_B6 * beta,
  i_E6 == i_B6 + i_C6,

  i_C11 == i_B11 * beta,
  i_E11 == i_B11 + i_C11,

  i_E10 == i_B11 + i_E6,

  i_E11 + i_E6 == i_in + i_out,

  i_B11 == i_B6,
  i_in == i_B10 + i_C11,
  i_out == i_C10

], [i_B10, i_B6, i_B11, i_C10, i_C6, i_C11, i_out, i_E10, i_E6, i_E11])

i[0][6].simplify_rational()

output

i_out == (beta^2 + 2*beta)*i_in/(beta^2 + 2*beta + 2)