# Linear Equations With One And Two Unkno

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• Published: November 14, 2014

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Course Notes Linear Math & Matrices
PSB – Dr. H. Schellinx

Linear
equations

As
we
have
seen,
a
linear
equation
with
n
different
variables,
say
x1, x2 , x3,..., xn ,
can

always
be
written
in
the
equivalent
standard
form
a1 x1 + a2 x2 + a3 x3 +... + an xn = c ,

where
c
is
a
constant,
the
xi
are
the
unknowns
and
the
ci
are
coefficients.

Here
are
some
examples: 5x − 3y = z + w − 6 is
a
linear
equation
with
4
unknowns
(x,y,w

and
z).
It
has
the
equivalent
standard
form
5x − 3y − w − z = −6 .

−3y = y + 2x − 7
is
a
linear
equation
with
2
unknowns
(x
and
y).
It
has
the
equivalent

standard
form
2x + 4y = 7 .

Linear
equations
with
one
unknown

Linear
equations
with
only
one
variable
are
the
simplest
ones.
We
can
always
write
such

an
equation
in
the
standard
form
ax = b .
Here
x
is
the
only
variable
(the
unknown).
The

coefficient
a
and
the
constant
b
are
both
real
numbers.

1. If
a ≠ 0 ,
the
equation
has
precisely
one
solution:
x =

b
.

a

2. If
a = 0 and
b ≠ 0 ,
the
equation
has
no
solution.
It
is
a

3. If
a = 0 and
b = 0 ,
all
real
numbers
satisfy
the
equation.
It
is
an
identity.

Recall
that
the
root
or
zero
of
a
function
f(x)
is
a
member
x
of
the
domain
of
f
such
that

f(x)
=
0
(we
say
that
f(x)
vanishes
at
x).

The
solution
of
the
linear
equation
with
one
unknown
ax = b ,
corresponds
to
the
root

of
the
linear
function
f (x) = ax − b .
In
the
two
dimensional
‘Euclidean
plane’,
using
a

rectangular
coordinate
system,
the
graph
of
this
function
is
the
straight
line
determined

by
the
equation
y = ax − b .

The
solution
of
the
equation
is
the
point
where
the
line
crosses
the
x-­‐axis
of
our

rectangular
coordinate
system.

As
an
example,
consider
the
following
linear
equation
with
one
unknown:
2x = 4 .
The

corresponding
linear
function
is
f (x) = 2x − 4 ,
which
has
as
its
graph
the
line

y = 2x − 4 .
The
solution
of
the
equation
is
the
zero
of
the
function,
i.e.
the
point
where

the
line
crosses