I have the following that I want to solve for v.

S = b - dv + r(b - dv) + c[b - dv + r(b - dv)] + t(b - dv + r(b - dv) + c[b - dv + r(b - dv)])

I started by getting rid of the parentheses:

S = b - dv + rb - rdv + cb - cdv + crb - crdv + tb - tdv + trb - trdv + tcb - tcdv + tcrb - tcrdv

Then, I reordered the terms and combined some like so:

S = b(1 + r + c + cr + t + tr + tc + tcr) - dv(1 - r - c - cr - t - tr - tc - tcr)

I have a few more steps that I took, but am wondering if someone could show me a better way to simplify and solve.

What I have so for is this, but I think I made a mistake along the way:

v = S - b[1 + r + c(1 + r + t + tr) + t + tr] / -d[1 - r - c(1 - r - t - tr) - t - tr]

Any guidance is appreciated.

Thanks,

Erik