Reaction forces on the top sphere

Begin by writing down the known and unknown quantities. The coordinates are for the centers of the spheres. The table is considered to be the $xy$ plane.

$R_5 = 5 \\ R_6 = 6 \\ R_7 = 7 \\ R_8 = 8 \\ (x_5, y_5, z_5) = (0,0,R_5) \\ (x_6, y_6, z_6) = (? ,0,R_6) \\ (x_7, y_7, z_7) = (? , ?,R_7) \\ (x_8, y_8, z_8) = (? ,?,?)$

Solve the following six nonlinear equations for $(x_6, x_7, y_7, x_8, y_8, z_8)$ .

$(x_5 - x_6)^2 + (y_5 - y_6)^2 + (z_5 - z_6)^2 = (R_5 + R_6)^2 \\ (x_5 - x_7)^2 + (y_5 - y_7)^2 + (z_5 - z_7)^2 = (R_5 + R_7)^2 \\ (x_5 - x_8)^2 + (y_5 - y_8)^2 + (z_5 - z_8)^2 = (R_5 + R_8)^2 \\ (x_6 - x_7)^2 + (y_6 - y_7)^2 + (z_6 - z_7)^2 = (R_6 + R_7)^2 \\ (x_6 - x_8)^2 + (y_6 - y_8)^2 + (z_6 - z_8)^2 = (R_6 + R_8)^2 \\ (x_7 - x_8)^2 + (y_7 - y_8)^2 + (z_7 - z_8)^2 = (R_7 + R_8)^2$

Having found the centers of all spheres, calculate unit vectors $\vec{u}_{5 8}$ , $\vec{u}_{6 8}$ , and $\vec{u}_{7 8}$ from the centers of spheres $5$ , $6$ , and $7$ to the center of sphere $8$ . These are the lines of action for the forces. Then solve the following linear equations for the forces:

$F_{58} u_{58x} + F_{68} u_{68x} + F_{78} u_{78x} = 0 \\ F_{58} u_{58y} + F_{68} u_{68y} + F_{78} u_{78y} = 0 \\ F_{58} u_{58z} + F_{68} u_{68z} + F_{78} u_{78z} = 100$

Results:

$(x_5, y_5, z_5) \approx (0,0,5) \\ (x_6, y_6, z_6) \approx (10.95 ,0, 6) \\ (x_7, y_7, z_7) \approx (4.2,11.06,7) \\ (x_8, y_8, z_8) \approx (3.14,0.507,17.6) \\ F_{58} \approx 72.74 \\ F_{68} \approx 30.76 \\ F_{78} \approx 5.61$