9/27/2004, LINEAR EQUATIONS
Math 21b, O. Knill
HOMEWORK due 9/29/2004: 1.1: 10,12,20,24,38b,26*,36* (Problems with * are recommended only).
SYSTEM OF LINEAR EQUATIONS. A collection of linear equations is called a
system of linear equations
.
An example is
3
x

y

z
= 0

x
+ 2
y

z
= 0

x

y
+ 3
z
= 9
.
It consists of three equations for three unknowns
x, y, z
.
Linear
means that no nonlinear terms like
x
2
, x
3
, xy, yz
3
,
sin(
x
) etc. appear. A formal definition of linearity will be given later.
LINEAR EQUATION. The equation
ax
+
by
=
c
is the general linear equation in two variables and
ax
+
by
+
cz
=
d
is the general linear equation in three variables. The general
linear equation
in
n
variables has the form
a
1
x
1
+
a
2
x
2
+
...
+
a
n
x
n
= 0
. Finitely many such equations form a
system of linear equations
.
SOLVING BY ELIMINATION.
Eliminate variables.
In the first example, the first equation gives
z
= 3
x

y
. Putting this into the second
and third equation gives

x
+ 2
y

(3
x

y
)
= 0

x

y
+ 3(3
x

y
)
= 9
or

4
x
+ 3
y
= 0
8
x

4
y
= 9
.
The first equation gives
y
= 4
/
3
x
and plugging this into the other equation gives 8
x

16
/
3
x
= 9 or 8
x
= 27
which means
x
= 27
/
8. The other values
y
= 9
/
2
, z
= 45
/
8 can now be obtained.
SOLVE BY SUITABLE SUBTRACTION.
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 Spring '03
 JUDSON
 Linear Algebra, Algebra, Linear Equations, Equations, Quadratic equation, Elementary algebra, Hyperplane

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