Friday, April 15, 2011

ax+by=cx+dy=zx+zy

Let ax+by=cx+dy=zx+zy;
Find z as function of a, b, c, and d;
Solution:

a(x/y)+b=c(x/y)+d=z(x/y)+z;
let (x/y)=t;
at+b=ct+d;
(a-c)t=d-b;
t=(d-b)/(a-c);
ct+d=zt+z;
z=(ct+d)/(t+1);
z={c(d-b)/(a-c)+d}/{(d-b)/(a-c)+1};
z={c(d-b)+d(a-c)}/{(d-b)+(a-c)};
z=(cd-cb+da-dc)/(d-b+a-c);
z=(cb+da)/(d-b+a-c);

z=(ad+bc)/(a-b-c+d);

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