Well, darn it all. I've got more than a dozen sheets of paper with over 100 equations, but I can't figure out why I'd scribbled:

ab + cd = -27

If we can justify that, then here's how it goes from there.

We add bc to each side:

ab + bc + cd = bc - 27

Now, we know that bc - 27 = -a - d, so we have:

ab + bc + cd = -a - d

And we know that da = 17 - b - c, so adding the respective sides gives:

ab + bc + cd + da = -a - d - b - c + 17

You'd written:

ab + bc + cd + da + 2(a + b + c + d) = 87

Therefore:

-a - b - c - d + 17 + 2(a + b + c + d) = 87

or

2(a + b + c + d) - (a + b + c + d) = 70

My brain is a bit fried; I'll try to revisit my papers later.