How to Solve Linear Equations
We give an algorithm to find out if a set of linear equations has a solution, and, if it
does have a solution, how to find
all
of the solutions.
Step One:
First we rewrite the set of linear equations dropping much of the redundant
information. If we have the equations
x

2
y
+ 3
z
= 4
y

7
z
= 14
2
x

z
= 0
,
then we write this as
1

2
3
4
0
1

7
14
2
0

1
0
.
Note that we separate out the coefficients from the constants. We call such an array an
augmented matrix. The sugmented part is the last column. A matrix is just a rectangular
array of numbers with none of the columns or rows set out. For example
3
4

6
23

7
0
is a matrix.
Step Two:
To (try) to solve this set of equations we perform special operations on the
equations, or rather we perform equivalent operations on the augmented matrix. These
operations do not change the set of solutions to our initial equations. We can do three
kinds of operations on the equations that do not change the set of solutions:
1. We can add a multiple of an equation to any other
different
equation.
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 Fall '08
 MARKMAN
 Linear Algebra, Algebra, Linear Equations, Equations, Row echelon form, augmented matrix

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