Linear Equations With One And Two Unkno

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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
 contradiction.
 
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
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