Mathl15
Notes
sohing a system
(2
by 2) of linear equations
AZby2 system
means
2 equations
in
2 unlpowrur. ln higher math
courses,
3
by 3 and
higher
systems are not
uncommon.
Knowledge
of matrices
(arrays)
& detenninarrs
(numbers
associated
with
these
matrices)
must be studied. In
this course, only 2 by
2
systems are considered.
There
are
various
methods
in algebra
used to solve these. However, one
method seems
to be the
most
convenient & easiest
to understand.
This method is called addition
& subtaction.
The
goal
of
all methods
is to eliminate
one ofthe variables
(symbols
forthe
quantities
appearing
in
both equations).
That way, the result
is one equation in the remaining
variable
& can
be solved
easily
forthat variable. Once that value is foun4
you
can use
it to find
the value of
the
other
variable.
In business applications, these systems arise mainly
from
supply
&
demand
equations.
Usually,
the two variables
represent
price
G)
tmd
quantity
of items
(q).
The solutions
will
give
what
we
call
the equilibrium
point
which
usually represents
where buyers
(consumers)
and sellers
(zuppliers)
settle
Let's
take the
example: 7p
+3q
=
686 and 5q
-
3p
:
1026
To use the addition & subtraction method
(called
just
addition
for short),
rewrite
the
system
so
that
the variables are
direcfly under each other.
We
would have:
3q+ 7p: 686
5q-- 3p
:
1026
Adding these will
not eliminate
a variable.
What we
need is a
----coeffiei-enf(number
h-fi6nt') nf-one ofTrenfrables
equal-fo-fheomeiffifia
fiffirEntEifu
-
-
It is
perrritted
to multiply
both sides of an equation
by any
number,.since
it
does not
change
the
solutions. We
can
achieve the
same
coefficients
with different signs
easily
by multiplying
the top
one by 5
and
the
bottom one by
-3.
This will
give
us 15q in
the top
& -l5q
in
the bottom.
We
could have multiplied
the
top
by
3
&
the bottom by
7 to
give
21p
in the
top
& -Zlp nthe
bottom.
This would
probably
be better
since
we would
not have
to multiply
by
a negative.
Let's
do that
This
gives:
9q+ ZLp=
2058
35q4lp
:7182
now,
just
add them.
We
get,
44q:9240,the
p's
are
eliminated.
This
gives q
:210.
(divide
both
sides by ag.
Using this
value for
q
and
substituting
into
the very
ls
equation, we
ge!
3(210)
*
7p
=
686,
which
glves
630
+
7p
:
686
or
7p: 56. Dividing
both
sides by
7, we
get,
p
=
8.
Therefore,
the simultaneous
solution
(works
in
both)
is
p8
&
q=2l}.This
would
represent
the
price
&
quantity
at the equilibrium
point
(where
buyers&
sellers
usuaily
settle).
