Expert: Paul Klarreich - 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
Questioner: H
Category: Advanced Math
Private: No
Subject: Re-arranging equations
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
...............................................
This looks like a pair of simultaneous equations in two variables -- no more.

Solve them for A and B:

c + A(d - c) = e + B(f - e)
w + A(x - w) = y + B(z - y)

A(d - c) = e + B(f - e) - c
A(x - w) = y + B(z - y) - w

A(d - c) - B(f - e) = (e - c)   (I)
A(x - w) - B(z - y) = (y - w)   (II)

Eliminate B: multiply I by (z - y), II by (f - e), then subtract.

A(d - c)(z - y) - B(f - e)(z - y) = (e - c)(z - y)   (I)
A(x - w)(f - e) - B(f - e)(z - y) = (y - w)(f - e)   (II)


 A(d - c)(z - y) - B(f - e)(z - y) =  (e - c)(z - y)   (I)
- A(x - w)(f - e) + B(f - e)(z - y) = -(y - w)(f - e)   (II)

I think you are on your way now.