a + b = 160
a + c = 150
b + d = 140
d + c = 130
a+b+0c+0d=160
a+0b+c+0d=150
0a+b+0c+1d=140
0a+0b+c+d=130
1 1 0 0 160
0 -1 1 0 -10
0 0 1 1 130
0 0 1 1 130
rank<4
it seems an infinite solution set
a + b = 160
a + c = 150
b + d = 140
d + c = 130
a+b+0c+0d=160
a+0b+c+0d=150
0a+b+0c+1d=140
0a+0b+c+d=130
1 1 0 0 160
0 -1 1 0 -10
0 0 1 1 130
0 0 1 1 130
rank<4
it seems an infinite solution set
I stopped there too. Multivariate equation is a dead end. Need to go back to fundamental geometic rules