Expert: Richard J. Raridon - 2/15/2010

Question
Hi, I have the following two equations:
c + A(d - c) = e + B(f - e)
w + A(x - w) = y + B(z - y)

And I need to re-arrange them to get the following two equations:

A = [(f - e)(w - y) - (z - y)(c - e)]/[(z - y)(d - c) - (f - e)(x - w)]

B = [(d - c)(w - y) - (x - w)(c - e)]/[(z - y)(d - c) - (f - e)(x - w)]

I have tried many times but still can't get the equations organised as they should be. Please could you show me the steps that I should take? Many Thanks

Answer
First of all, I would re-write them A(d-c) = e-c +B(f-e) and A(x-w) = y-w +B(z-y)
Multiply each term in the 1st eq.by (z-y) and each term in the 2nd eq. by (f-e) to get
A(d-c)(z-y) = (e-c)(z-y) +B(f-e)(z-y)
A(x-w)(f-e) = (y-w)(f-e) +B(f-e)(z-y)
now you can subtract the 2 eqns. to get A[(d-c)(z-y)-(x-w)(f-e)] = (e-c)(z-y)-(y-w)(f-e)
Solve for A and change a couple of minus signs to get what you want.  
Now do the same thing and eliminate A and solve for B.