Linear Systems

Date: 10/22/96 at 20:51:13
From: Brett Farley
Subject: Linear systems
 
I need to solve this system of equations for x and y in terms of a, b, 
c, d, e, and f:

ax + by = e
cx + dy = f

I've come up with:  (x,y) = de - bf    ce - af
                            _______  , _______
                            da - bc    ab - ad 

Date: 10/23/96 at 13:22:35
From: Doctor Ken
Subject: Re: Linear systems

Hi Brett-

Your answer isn't quite right, but it's close.

The correct answer should have been

          / d e - b f    f a - c e \
 (x, y) = | ---------,   --------- |
          \ d a - b c    d a - b c /

x was right, and the numerator for y was right.  So you should try to 
figure out where y got messed up in your equations, and then you'll be 
all set.  

If you go on and study more math, you'll learn some methods that make 
this problem much easier.  Those methods, studied in a class called 
"Linear Algebra," have to do with matrices, which are just tables of 
numbers.  If you're interested in knowing about these, read on!

A matrix is just a bunch of rows and colums.  In your case, we could 
write the problem as

/ a  b \/ x \    / e \
|      ||   |  = |   |
\ c  d /\ y /    \ f /

That's a 2 by 2 matrix times a 2 by 1 matrix, and on the right is a 2 
by 1 matrix.  

Anyway, solving that problem for x and y isn't bad if you know the 
techniques.  But if you don't use matrices and just try to use regular 
high-school algebra techniques, you're likely to take longer and make 
more mistakes. 

-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/