with numeric unput

Clear[x1]; Clear[x2]; Clear[x3]; Clear[x4]; Clear[x5]; Clear[x6];
Clear[x7]; Clear[x8]; Clear[x9]; Clear[x10]
Clear[y1]; Clear[y2]; Clear[y3]; Clear[y4]; Clear[y5]; Clear[y6];
Clear[y7]; Clear[y8]; Clear[y9]; Clear[y10];
u1 = 60.5 ; u2 = 71; u3 = 172; u4 = 178; u5 = 21.5; u6 = 35.9; u7 =
130; u8 = 139; u9 = 148; u10 = 157;
v1 = 102; v2 = 62.9; v3 = 80.4; v4 = 37.1; v5 = 250; v6 = 197; v7 =
186; v8 = 133; v9 = 243; v10 = 183;
NSolve[u1 == (m13 + m11 x1 + m12 y1)/(1 + m31 x1 + m32 y1) &&
  v1 == (m23 + m21 x1 + m22 y1)/(1 + m31 x1 + m32 y1) &&
  u2 == (m13 + m11 x2 + m12 y2)/(1 + m31 x2 + m32 y2) &&
  v2 == (m23 + m21 x2 + m22 y2)/(1 + m31 x2 + m32 y2) &&
  u3 == (m13 + m11 x3 + m12 y3)/(1 + m31 x3 + m32 y3) &&
  v3 == (m23 + m21 x3 + m22 y3)/(1 + m31 x3 + m32 y3) &&
  u4 == (m13 + m11 x4 + m12 y4)/(1 + m31 x4 + m32 y4) &&
  v4 == (m23 + m21 x4 + m22 y4)/(1 + m31 x4 + m32 y4) &&
  u5 == (m13 + m11 x5 + m12 y5)/(1 + m31 x5 + m32 y5) &&
  v5 == (m23 + m21 x5 + m22 y5)/(1 + m31 x5 + m32 y5) &&
  u6 == (m13 + m11 x6 + m12 y6)/(1 + m31 x6 + m32 y6) &&
  v6 == (m23 + m21 x6 + m22 y6)/(1 + m31 x6 + m32 y6) &&
  u7 == (m13 + m11 x7 + m12 y7)/(1 + m31 x7 + m32 y7) &&
  v7 == (m23 + m21 x7 + m22 y7)/(1 + m31 x7 + m32 y7) &&
  u8 == (m13 + m11 x8 + m12 y8)/(1 + m31 x8 + m32 y8) &&
  v8 == (m23 + m21 x8 + m22 y8)/(1 + m31 x8 + m32 y8) &&
  u9 == (m13 + m11 x9 + m12 y9)/(1 + m31 x9 + m32 y9) &&
  v9 == (m23 + m21 x9 + m22 y9)/(1 + m31 x9 + m32 y9) &&
  u10 == (m13 + m11 x10 + m12 y10)/(1 + m31 x10 + m32 y10) &&
  v10 == (m23 + m21 x10 + m22 y10)/(1 + m31 x10 + m32 y10) &&
  dx == x2 - x1 && dx == x4 - x3 && dx == x6 - x5 && dx == x8 - x7 &&
  dx == x10 - x9 && dy == y2 - y1 && dy == y4 - y3 && dy == y6 - y5 &&
  dy == y8 - y7 && dy == y10 - y9
 , {dx, dy, m11, m12, m13, m21, m22, m23, m31, m32,
  x1, x2, x3, x4, x5, x6, x7, x8, x9, x10,
  y1, y2, y3, y4, y5, y6, y7, y8, y9, y10}]

without division
Clear[x1]; Clear[x2]; Clear[x3]; Clear[x4]; Clear[x5]; Clear[x6];
Clear[x7]; Clear[x8]; Clear[x9]; Clear[x10]
Clear[y1]; Clear[y2]; Clear[y3]; Clear[y4]; Clear[y5]; Clear[y6];
Clear[y7]; Clear[y8]; Clear[y9]; Clear[y10];
u1 = 60.5 ; u2 = 71; u3 = 172; u4 = 178; u5 = 21.5; u6 = 35.9; u7 =
130; u8 = 139; u9 = 148; u10 = 157;
v1 = 102; v2 = 62.9; v3 = 80.4; v4 = 37.1; v5 = 250; v6 = 197; v7 =
186; v8 = 133; v9 = 243; v10 = 183;
NSolve[u1 (1 + m31 x1 + m32 y1) == (m13 + m11 x1 + m12 y1) &&
  v1 (1 + m31 x1 + m32 y1) == (m23 + m21 x1 + m22 y1) &&
  u2 (1 + m31 x2 + m32 y2) == (m13 + m11 x2 + m12 y2) &&
  v2 (1 + m31 x2 + m32 y2) == (m23 + m21 x2 + m22 y2) &&
  u3 (1 + m31 x3 + m32 y3) == (m13 + m11 x3 + m12 y3) &&
  v3 (1 + m31 x3 + m32 y3) == (m23 + m21 x3 + m22 y3) &&
  u4 (1 + m31 x4 + m32 y4) == (m13 + m11 x4 + m12 y4) &&
  v4 (1 + m31 x4 + m32 y4) == (m23 + m21 x4 + m22 y4) &&
  u5 (1 + m31 x5 + m32 y5) == (m13 + m11 x5 + m12 y5) &&
  v5 (1 + m31 x5 + m32 y5) == (m23 + m21 x5 + m22 y5) &&
  u6 (1 + m31 x6 + m32 y6) == (m13 + m11 x6 + m12 y6) &&
  v6 (1 + m31 x6 + m32 y6) == (m23 + m21 x6 + m22 y6) &&
  u7 (1 + m31 x7 + m32 y7) == (m13 + m11 x7 + m12 y7) &&
  v7 (1 + m31 x7 + m32 y7) == (m23 + m21 x7 + m22 y7) &&
  u8 (1 + m31 x8 + m32 y8) == (m13 + m11 x8 + m12 y8) &&
  v8 (1 + m31 x8 + m32 y8) == (m23 + m21 x8 + m22 y8) &&
  u9 (1 + m31 x9 + m32 y9) == (m13 + m11 x9 + m12 y9) &&
  v9 (1 + m31 x9 + m32 y9) == (m23 + m21 x9 + m22 y9) &&
  u10 (1 + m31 x10 + m32 y10) == (m13 + m11 x10 + m12 y10) &&
  v10 (1 + m31 x10 + m32 y10) == (m23 + m21 x10 + m22 y10) &&
  dx == x2 - x1 && dx == x4 - x3 && dx == x6 - x5 && dx == x8 - x7 &&
  dx == x10 - x9 && dy == y2 - y1 && dy == y4 - y3 && dy == y6 - y5 &&
  dy == y8 - y7 && dy == y10 - y9
 , {dx, dy, m11, m12, m13, m21, m22, m23, m31, m32,
  x1, x2, x3, x4, x5, x6, x7, x8, x9, x10,
  y1, y2, y3, y4, y5, y6, y7, y8, y9, y10}]


affine m31=m32=0
no solutions?
Clear[x1]; Clear[x2]; Clear[x3]; Clear[x4]; Clear[x5]; Clear[x6];
Clear[x7]; Clear[x8]; Clear[x9]; Clear[x10]
Clear[y1]; Clear[y2]; Clear[y3]; Clear[y4]; Clear[y5]; Clear[y6];
Clear[y7]; Clear[y8]; Clear[y9]; Clear[y10];
u1 = 60.5 ; u2 = 71; u3 = 172; u4 = 178; u5 = 21.5; u6 = 35.9; u7 =
130; u8 = 139; u9 = 148; u10 = 157;
v1 = 102; v2 = 62.9; v3 = 80.4; v4 = 37.1; v5 = 250; v6 = 197; v7 =
186; v8 = 133; v9 = 243; v10 = 183;
m32 = 0; m31 = 0;
NSolve[u1 (1 + m31 x1 + m32 y1) == (m13 + m11 x1 + m12 y1) &&
  v1 (1 + m31 x1 + m32 y1) == (m23 + m21 x1 + m22 y1) &&
  u2 (1 + m31 x2 + m32 y2) == (m13 + m11 x2 + m12 y2) &&
  v2 (1 + m31 x2 + m32 y2) == (m23 + m21 x2 + m22 y2) &&
  u3 (1 + m31 x3 + m32 y3) == (m13 + m11 x3 + m12 y3) &&
  v3 (1 + m31 x3 + m32 y3) == (m23 + m21 x3 + m22 y3) &&
  u4 (1 + m31 x4 + m32 y4) == (m13 + m11 x4 + m12 y4) &&
  v4 (1 + m31 x4 + m32 y4) == (m23 + m21 x4 + m22 y4) &&
  u5 (1 + m31 x5 + m32 y5) == (m13 + m11 x5 + m12 y5) &&
  v5 (1 + m31 x5 + m32 y5) == (m23 + m21 x5 + m22 y5) &&
  u6 (1 + m31 x6 + m32 y6) == (m13 + m11 x6 + m12 y6) &&
  v6 (1 + m31 x6 + m32 y6) == (m23 + m21 x6 + m22 y6) &&
  u7 (1 + m31 x7 + m32 y7) == (m13 + m11 x7 + m12 y7) &&
  v7 (1 + m31 x7 + m32 y7) == (m23 + m21 x7 + m22 y7) &&
  u8 (1 + m31 x8 + m32 y8) == (m13 + m11 x8 + m12 y8) &&
  v8 (1 + m31 x8 + m32 y8) == (m23 + m21 x8 + m22 y8) &&
  u9 (1 + m31 x9 + m32 y9) == (m13 + m11 x9 + m12 y9) &&
  v9 (1 + m31 x9 + m32 y9) == (m23 + m21 x9 + m22 y9) &&
  u10 (1 + m31 x10 + m32 y10) == (m13 + m11 x10 + m12 y10) &&
  v10 (1 + m31 x10 + m32 y10) == (m23 + m21 x10 + m22 y10) &&
  dx == x2 - x1 && dx == x4 - x3 && dx == x6 - x5 && dx == x8 - x7 &&
  dx == x10 - x9 && dy == y2 - y1 && dy == y4 - y3 && dy == y6 - y5
 , {dx, dy, m11, m12, m13, m21, m22, m23,
  x1, x2, x3, x4, x5, x6, x7, x8, x9, x10,
  y1, y2, y3, y4, y5, y6, y7, y8, y9, y10}]